Hm… I’m gonna try seeing this from Mafia’s POV.
Two checks (red and green)
Correct Red & Correct Green → Maybe Cop
Correct Red & Incorrect Green → Maybe Cop redcheck
Incorrect Red & Correct Green → Maybe Cop greencheck
Incorrect Red & Incorrect Green → Definitely not Cop
One check (green-only)
Correct Green → Maybe Cop
Incorrect Green → Maybe Cop (because maybe redcheck)
One check (red-only)
Correct Red → Maybe Cop
Incorrect Red → Maybe Cop (because maybe greencheck)
One check (random)
Correct check → Maybe Cop
Incorrect check → Unlikely Cop (because why would Cop give false result?)
As far as security goes, two-checks is less secure than one-check, because if someone gives two wrong results, then Mafia knows they are not Cop. In this aspect (i.e., security), one predetermined check gives the least amount of information to Mafia.
However, as far as efficiency goes (i.e., Cop giving correct info to Town), one-check-random gives the best result. Except this is such a terrible method, since Mafia would immediately know that anyone who gives wrong checks are not Cop.
. . .
Okay, maybe I should stop thinking about this. Obviously Town would have to choose which option is the lesser evil, but at the moment I am starting to believe Frost might be right about how discussing this might not be a good idea at all.