The murderer is is Karube Teigo, the architect. He used his collapsible umbrella to murder Hoshina Rukako. The handle part concealed the icepick-like blade, while the umbrella part was used to block the majority of the blood from splashing onto his clothes. After reassembling and collapsing the umbrella, he disposed of the umbrella down the well by using some contraption tied to the well and the tower clock.
Why murder didn't occur until the third year
The first clue is that the murderer sent the letter twice previously, but there is no indication of a countdown. This mean the circumstances weren’t correct for the murderer on Hoshino’s last two birthdays. It is raining this time which provides the excuse to have an umbrella. Karube Teigo has a foldable umbrella he kept with him and he even put it in his jacket while wet instead of checking it with the butler. Also the rain would hide string contraptions more effectively because people would be less likely to look up.
Timeline of events
After everyone cleared out of the clocktower, but before the party, Karube Teigo took the time to prepare the contraption/string trick at the well and the tower clock. The contraption works so when the tower clock strikes six, a string is cut or slips which causes the string that had been run through the well grating and then the balcony door to pull the umbrella that the culprit tied to it into the well. I imagine there is some weight dangling in the well which will cause the umbrella to be yanked at high speed, enough to throw open the unlocked, outward-facing door. I'll try to elaborate on the mechanics later.
During the party, Karube Teigo took the time to touch Hoshina Rukako's necklace and apply luminescent paint. He also asks to examine Hoshina's pocketwatch. He had to set Hoshina's pocketwatch fast to ensure he has time between when she blows out the candles and the tower clock strikes six to attach the murderous umbrella to the contraption before it sets off. The butler says that Hoshina always times the blowing out of the candles with the first chime of the clocks. Ran points out that Hoshina was early, and the butler says the clock technicians assigned said the clocks went off at the correct time. This means Hoshina's pocketwatch, which was her reference, was fast. It is most likely the reason she was angry right before she was murdered because she noticed the clocks didn't go off properly according to her.
When the butler turns the light out, Karube Teigo opens the umbrella and separates the handle which conceals an icepick like blade. When he does this, it creates a cloth friction sound like a lady's dress moving which Karube himself noted. In the process of getting close to Hoshina Rukako, the umbrella brushes up against Furugaki Rinsaku who mistook the springy feeling for a fat person's body passing him. The umbrella seems like a nimble person to Suou Chiaki because it is lightweight and easily moves.
I hazard that Karube Teigo held the umbrella near his body with one hand to block the blood spatter and held the separated handle blade with the other, and he stabbed the blade through the umbrella into Hoshina Rukako's back. This resulted in the unusual parabola shaped boundary where the blood was blocked by the circular shape of the umbrella. He then closed the umbrella and reattached the handle which would get blood on his hands and cuffs, tied it to the contraption, and then positioned himself to kneel when the lights are turned on so he could explain the blood on his hands and cuffs.
The contraption goes off right when the tower clock strikes six which pulls the umbrella through the door at high speed, flinging it open and leaving rubbed blood marks on porch from dragging. The umbrella is dragged through the grating into the well and the left over string either goes with it or is wrapped up into the gears of the tower clock somehow. The weather forecast said the rain should have stopped in the evening. Karube Teigo was counting on that to avoid questions about the absense of footprints, but it didn't turn out the way he wanted casting suspicion on the party members.
Evidence
There should be plenty of evidence such as fingerprints on the umbrella since Karube was not wearing gloves earlier. Karube also doesn't have the umbrella when he hits questioning despite putting it in his jacket earlier. Any remnants of string in the tower clock and the well and the pocket watch being fast will prove the mechanics of the trick. Karube hasn't washed his hands which have blood on them before he was questioned, so if he applied the luminescent paint with his bare fingers, the paint should show up among the blood with a black light; however, other people handling the necklace later might have paint on them as well. The same goes for Karube's fingerprints being on the pocketwatch because other people might have handled it.
I still have to explain the positioning of the other suspects and why they have the bloodstains on certain parts of their body, but I imagine the umbrella probably protected them as well from most of the the blood splatter. I also need to prove that no one else could have committed the crime, but I think Karube is the only one who handled the pocketwatch before the crime with an umbrella and could set the previously precisely ontime pocketwatch fast. The mysterious bit is why Karube pressed for an investigation when he bothered to set up the trick in the first place. He did say that he wasn't keen that the ghost of his childhood friend could be made out to be the culprit. If he cared enough to murder someone, then perhaps he cares enough not to sully the name of his dead friend?
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