Bidding order is not something that they could safely fake even later in the game
This is not technically true; they'll know each other's bidding orders by now, at least within the mafia faction. Plus, I'm assuming bidding order doesn't flip with a player, meaning with the info scum has, they're more likely to be able to infer a named PR's bidding order than we are.
plus it's not even alignment-indicative, so I don't really know what the benefit of this would be.
It adds a load of constraints about who could have claimed what slot, and those constraints get tighter as the game goes on and people flip. The cool thing is that scum probably went for scum-beneficial PRs, so if there's a clear towny PR in a slot we know they must have selected, and they can't prove they have that PR, we've got them cornered. So we get to hold the threat of slot-claiming over scum, and in terms of adding setup constraints to boost our ability to solve things later on, I think it's a clear benefit. The question for me is just whether that benefit outweighs the cost of outing a few of the highest-ranked people as possible/probable PRs. I don't see why the fausts and Cals of this game aren't at least discussing the idea.
(I do get why the 2.7s of the game aren't... e and I have a big TvT spat about this sort of constraint-driven approach in RMM37 and we lost, so I'm not saying it's a clear strategy, just one that I'm surprised people aren't at least looking at).