Frank Stillwell states:
“After Frank joins the mark’s posse, or the posse opposing a job, if he is wanted, he unboots.”
Checking the rules on when you receive bounty for fightin’ in someone else’s private place, it says:
"When a shootout takes place at a Private location, regardless of who started it, all dudes in the shootout that are not controlled by the owner of that location have their bounty increased by 1. If this takes them from 0 to 1 bounty, they are now wanted. "
It isn’t clear exactly when this happens. As I understand it, as soon as a Dude is in a Shootout in a Private location, bounty is applied. Thus, if Frank Stillwell was opposing a Job at an opponent’s Home, order of operations would be:
a) Frank Stillwell moves to opponent’s Home.
b) Frank Stillwell joins posse.
c) Frank Stillwell gains bounty.
d) After posses are formed, Frank Stillwell unboots.
I could see the argument, though, that where the rules on bounty placement come after the rules on posse formation, bounty is placed on dudes at a Private location after posses are formed. Still, that timing would be the same as Frank’s trait, and would also seem a little janky with respect to movement mid-shootout.
I feel like the timing on Bounty placement is something that might already be flagged for clarification somewhere, but if so I’m overlookin’ where in the Rules it was clarified.
Thanks, as always, fer yer diligent efforts to herd us cats back to the light of the Glossary! 
(ST:CCG joke).