If that's the case, then, again, Gosho could just have his assistants do it.
I still think he's more likely to be Rum than Rumi, but weighed against Kanenori and Hyōe? Right now, I'll take them over Chikara. While Muga's involvement has been fairly consistent since his debut, Chikara's is more lengthly—in fact, he was introduced (899–902/785–786, at the beginning of the arc) even before Hyōe (913–917/810–812, around 15 files into the arc). After the Bourbon arc (where Rei first appeared as Scar Akai in 677–679/563–564—50 files into the arc, after Shūichi/Subaru—and then appeared as himself in 793–795/667–668, 170 files into the arc, after Masumi), we'd expect Rumi (introduced in 966–968/889–890, nearly 70 files into the arc), Kanenori (975–977/894–895, nearly 80 files into the arc) or Muga (introduced in 981–983/885–886, over 80 files into the arc) to be Rum, in terms of suspect introduction order... so I'd say it'd be more of a subversion, in those terms, if Hyōe or Chikara turned out to be Rum.
In terms of Rum arc appearances...
Chikara has had 2 cameo appearances (899–902/785–786 and 945–947/849–850).
Hyōe has had 6 cameo appearances (918–920/814–815; 978–980/896–897; 1,000–1,005/927–928; 1,006–1,008/941–942; 1,013–1,017; and 1,027–1,031), and 2 appearances where he was in the thick of the case from start to finish (913–917/810–812 and 987–989/909–910).
Rumi has had 2 cameo appearances (1,000–1,005/927–928 and 1,006–1,008/941–942), and 4 appearances where she was in the thick of the case from start to finish (966–968/889–890; 978–980/896–897; 987–989/909–910 and 1,032–1,034).
Kanenori has had 3 cameo appearances (978–980/896–897; 1,000–1,005/927–928 and 1,006–1,008/941–942), and 2 appearances where he was in the thick of the case from start to finish (975–977/894–895 and 1,027–1,031).
Muga has had 2 cameo appearances (990–993/916–917 and 1,000–1,005/927–928), and 2 appearances where he was in the thick of the case from start to finish (981–983/885–886 and 1,039–1,042).
For a point of comparison, here are Bourbon arc appearances...
Shūichi/Subaru had 5 cameo appearances (635–637/524–525; 690–692/571–572; 753–755/622–623; 775–777/656–657 and 815–817/699–700), and 8 appearances where he was in the thick of the case from start to finish (622–624/509–510; 638–640/510–511; 664–666/542–543; 700–704/578–581; 796–800/671–674; 809–811/684–685; 812–814/690–691 and 818–824/701–704).
Masumi had 4 cameo appearances (775–777/656–657; 781–786/652–655; 787–789/661–662 and 801–803/675–676), and had 7 appearances where she was in the thick of the case from start to finish (768–770/646–647; 771–774/648–650; 778–780/651; 796–800/671–674; 812–814/690–691; 815–817/699–700 and 818–824/701–704).
Rei/Bourbon had 3 cameo appearances (801–803/675–676; 804–808/681–683 and 815 and 817/699–700), and had 5 appearances where he was in the thick of the case from start to finish (677–679/563–564 and 700–704/578–581 as Scar Akai; 793–795/667–668, 796–800/671–674 and 818–824/701–704 as Tōru Amuro).
And for another point of comparison, here are Vermouth arc appearances...
Vermouth as Dr. Araide had 3 cameo appearances (254–257/190–191; 355–357/274–275 and 420–422/338–339), and had 3 appearances where she was in the thick of the case from start to finish (287–289/230–231; 328–330/253–254; and 429–434/345).
Jodie had 7 appearances where she was in the thick of the case from start to finish (270–272/226–227; 287–289/230–231; 338–340/271–272; 341–343/277–278; 420–422/338–339; 426–428/343–344 and 429–434/345).
Shūichi had 8 cameo appearances (325–327/258–259; 328–330/253–254; 335–337/269–270; 350–354/286–288; 377–379/307–308; 380–383/309–311; 417–419/335–336 and 429–434/345), and had 2 appearances where he was in the thick of the case from start to finish (287–289/230–231 and 420–422/338–339).
Just putting that info out there.
Certainly, it'd lend to Chikara's involvement if he had more than cameo appearances.
Right now, I think it's most likely Chikara will have some other involvement—right now, I think that's more likely than him being a BO member.
As for Rei's "impatient" hint... is he solely basing that on the texts he's gotten from Rum to hurry up with his Shinichi investigation? If so, then I wouldn't count it as a point against Chikara being Rum. If not, then I'd agree with both of you that it is a point against Chikara being Rum.
