Ichimaru Gin's Psychologist claim
All analysis is done with the assumption that the claim is a lie. We are trying to find sensibility in the choice of fake claim. We will do this for all four claimants.
Assumption #1: a fakeclaim would be for survival, not drawing out a counterclaim.
Assumption #2: therefore, a fakeclaim would be a role that could not be counterclaimed.
1. Detective, 1-shot Commuter, Goon Cop; 1-shot Investigation Immune + Serial Killer
2. Watcher, Jailkeeper, Psychologist; 2-shot JOAT
3. 1-shot BP Townie, Jailkeeper, Bodyguard; Traitor Knows Mafia
4. 1-shot BP Commuter, Psychologist, Detective; Mafia Knows Traitor
5. Watcher, 1-shot BP Townie, 1-shot BP Commuter; 2-shot JOAT
6. Detective, Jailkeeper, Psychologist; Mafia Knows Traitor + Serial Killer
7. 1-shot Commuter, Jailkeeper, Detective; Traitor Knows Mafia + Serial Killer
8. Goon Cop, Psychologist, Bodyguard; 1-shot Investigation Immune
The green setups are ones in which a Psychologist exists. We have to assume that Ichimaru Gin knew the game we are playing is NOT one of these. All "Mafia Knows Traitor (MKT)" setups are in green. In #3 and #7, "Traitor Knows Mafia (TKM)."
If Ichimaru Gin is mafia, and he decided to fakeclaim Psychologist, it appears most sensible that he would do this as a Traitor in a TKM setup, because in all cases, he knows Psychologist is a safeclaim. As a Mafia Goon in a MTK setup, there is a 100% chance that a real Psychologist is in the game, so he doesn't fakeclaim Psychologist. As a Mafia Goon that does NOT know the Traitor, that's six possible setups, with him knowing he's in a set of two {1 or 8}, {2 or 5}, or {3 or 7}. If he's in the third set, he knows Psychologist is a safeclaim; in the other two sets, it's a 50% chance, so there are better choices. If he's a Traitor who does NOT know Mafia, he is basically playing blind, as he has no setup information. In that case, VT has to be the safest claim.
If Ichimaru Gin is a Serial Killer, he knows the setup is one of {1, 4, 6, 8}. That's a 75% chance a Psychologist exists.
So, if Ichimaru Gin is...
a Mafia Goon in an MTK setup -- he does NOT fakeclaim Psychologist
a Mafia Goon in an non-MTK setup -- he has only a 33% chance of being in a setup where Psychologist is a safe fakeclaim
a Mafia Traitor in a TKM setup -- Psychologist is a safe fakeclaim
a Mafia Taritor in a non-TKM setup -- he would have to claim VT
a Serial Killer -- he has only a 25% chance of being in a setup where Psychologist is a safe fakeclaim
Based on this, I believe there is only one scenario where Ichimaru Gin fakeclaims Psychologist, and that is as the Mafia Traitor in a TKM setup.