I'm really late with an attempt at the explanation this time, but real life got in the way. I feel like I did a pretty crummy job with this so please think of contradictions, details I missed, and logical trails which I may have not completely followed to their conclusions. (despoilered for searching purposes...)
Only one person crossed the bridge in the morning at 4:43 AM. The time of death was around 5AM (probably earlier). If the heat was going to throw off the time of death, it would make it seem like it occurred earlier than it did. Given that the timeframe is probably close to exact, I doubt the heat plays a role. Conan and friends uncovered the body sometime between ten till five and five sharp. There would be little time to dispose of much evidence.
The other party in this case would have had to camp out all night behind the bath. Conan didn't hear anyone jump into the lake, but would he have heard the sound of someone throwing something into the lake which would have made a quieter splash?
Regarding room keys, someone planning to wait out all night would probably keep their room key with them. It would seem strange to leave the key with reception late at night, and the reception would remember if that person didn't pick up the key again, yet didn't check out.
The person who waited out
There are several facts which collectively hint that Tetsuyama Ganji was the one who waited out.
Tetsuyama Ganji was already naked over by the bath. Someone with muscle pain to the point of having difficulty walking would probably take their time removing their clothes. Also, normally you are supposed to shower thoroughly before you enter the hot spring. None of the shower heads are on.
For the floating match sticks, two people are suspect: Teradou Unpei and the screenwriter, Tetsuyama Ganji, who uses a pipe. Pipe smokers tend to prefer matches although some use lighters. There isn't any evidence like burns or smoky smells that indicate that something large was burnt, so the matches were probably used for smoking. Teradou Unpei is probably a heavy smoker given the state of his ashtray. If cigarette butts float for at least a short period of time, then their absence would appear to implicate Tetsuyama Ganji as the person who waited out.
Edit: The tea bottle filled with water may have been Tetsuyama Ganji's. Remember Agasa'a bath gadget commented on the health benefits of drinking the water. Perhaps Tetsuyama Ganji' who had injured his muscle (very vague injury), was intending to reap those benefits? This may also help explain why he waited out or at least wanted to get to the onsen first: he wanted clean water that no one had bathed in?
Finally there is Tetsuyama Ganji's strange appearance. He was wearing his towel as a muffler, and did not look particularly excited as expected of someone going to do something perverted. The screenwriter intended to meet the actress Natori Mishio which would explain why he would choose to go to the women's bath. However, all the suspects likely knew of the plan, Tanzawa Junsaku being Natori's manager and Teradou Unpei from eavesdropping. I think it is likely one of the other suspects disguised as him instead knowing he was already waiting over at the baths.
As for why Tetsuyama decided to wait out, most likely it is for a perverted reason, but if someone took his room key while he was bathing so they had access to his room to get materials necessary for disguise, then it is possible that he may have noticed his key was missing and decided to wait out rather than get reception to break him in. Maybe the killer left him a note pretending to be Natori Mishio agreeing to sneak out sometime during the night and to wait. I'm not really sure.
Stuff the killer would have had to worry about which create potential clues.
The killer who disguised as Tetsuyama Ganji would need several articles. He needed a towel as a muffler, a beanie hat, spare glasses, and a heavy jacket as described by the people who let him across. The killer would be wearing his normal clothing underneath. If he was carrying a murder weapon it would have to be something which could be hidden under his clothes.
The murderer may also be wearing the screenwriter's ring for added convincing effect. The screenwriter did not seem to be wearing it in chapter 722 even though it was mentioned he liked showing it off.
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When the killer arrived at the bath where Tetsuyama was probably nude and waiting thinking Natori was coming, the killer likely used one of the nearby bathing stones or something they brought with them as the weapon. After killing the screenwriter, murderer positioned the body so it appeared to be hit by a rock, applied some soap to the screenwriter's heel, and then left the soap nearby.
The killer would have had to touch the soap. Soap does take fingerprints if it is wet and soft. It may also leave residue on killer's hands. The police should examine bar of soap for fingerprints, and the suspects for soap residue, especially on their heels. Also soap will generally deform if stepped on and leave scrapings on the ground. If the killer intended to mimic these scene accurately the soap bar should have gouges and possibly also the killer's heelprint.
The killer then probably used the water filled tea bottle which may have been prepared ahead of time and smuggled across the bridge or left by Tetsuyama who was waiting out to dump water on the victim to wash away unnatural bloodstains. The murderer may have also used the water bottle to clean themselves off as well. I feel like I haven't thought enough about the tea bottle. Edit: I now think the tea bottle was Genji's. See above for details.
The murderer would have had to change out of the top layer of his clothes, and dispose of the older set of clothes belonging to Tetsuyama who waited out the night as it would expose the trick if two sets of clothes were left. The likely location of the clothes is in the lake nearby the bath on the veranda side. Something heavy was probably used to sink the clothes quickly, perhaps a bathing rock or the murder weapon.
The murderer would have probably needed a towel or something in order to keep their fingerprints from being on various items. This towel may have blood on it so it may have been sunken along with the older set of clothes.
Given the short time frame available to cleanup, the murderer may not have had time to rifle through the pockets to find everything, so there may be personally identifying objects like the screenwriter's room key, or his pipe, or tobacco supplies which would identify the clothing as Tetsuyama's. Also, if the screenwriter was waiting outside and then came in through the veranda entrance to avoid being seen, he would have left fingerprints on the sliding door and probably around where he was sitting down waiting as well.
All the suspects are professional actors and thus I am going to assume have adequate disguising and acting skills to impersonate the victim.
Natori Mishio
She gave her room key to the people at the front desk. If she woke up when she claimed to, then she probably dropped off her key between five and five thirty. This would seem to exonerate her because she was on the wrong side of the bridge when only one person, someone who looked like the victim, could cross over.
She probably didn't drop off her key early, disguise as the victim, and then cross over because the reception would remember her since the baths aren't open yet. Many hotel receptions have cameras mounted by them, so there is a chance to check her alibi. It might be possible to drop off her key while dressed as the victim to make it seem like she left her key later.
Natori Mishio would have the hardest time disguising as the victim because she would need padding and potentially also a mask. A makeup kit and towel would be necessary to remove the makeup she was wearing if she disguised as the screenwriter. Does the makeup kit include a mirror I wonder? She would probably need one to make sure she removed the makeup thoroughly which would be time consuming. She would need to dispose of the towel if she did remove her makeup because the makeup left on her towel would be suspicious. She may have brought multiple towels though. The towel could also be used to keep her from leaving fingerprints on something.
Also Tanzawa Junsaku noted she had been missing from her room for a long time, but the timeframe she was missing isn't clear.
Teradou Unpei
It is strange that he brought cigarettes yet didn't bring matches to go with them. Also he shows off the items he is holding with one hand, his right. I don't have much to say about him really. It would be easier for a man to disguise as the victim.
Tanzawa Junsaku
He is the only suspect where it is definitely known to be on the hotel side after the baths were closed for the evening. As with Unpei, it would be easier for a man to disguise as the victim.
Despite what he said about being right handed, Tanzawa Junsaku seems to be left handed based on his interaction with the vending machine. The golf glove in his possession would cover up his dominant hand, but it was found in his room. I'm not sure if he would have had time to go back and do something with it. In any case, he may have used a towel instead to prevent fingerprints.
Getting change for the drink machines from the hotel reception may be a way of confirming his alibi.
Incidentally
he is wearing a ring on his left hand middle finger (when showing what he had in his pockets) at the crime while
he was not wearing one thirty minutes earlier when purchasing drinks. There isn't a stone visible, but he may have turned the ring around so the stone doesn't show. Also, if he is left handed, he may have touched the bloody body with his left hand to position it better. Blood may get stuck under the ring so even if he washed his hands, there may be a significant luminol reaction from the inner side of the band or his finger where it covers.
He also attempted to get close to the corpse. Conan held him back, but he may have done so to offload a critical piece of evidence, like the ring?
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