In this scenario though, we can keep looking for scum outside of Joseph and I WITHOUT giving up the ability to demonstrably prove whether or not we told the truth. I could see this being a potential fakeclaim later in the day when choices have to be made, since in a lot of the scenarios, the exile goes through on the 3rd party if the drunkmayor is lying, but D1 where we can create 2 ICs (or 50/50 a scum), and still have time to individually test both of us on future wagons without sacrificing other scumhunting makes this a terrible fakeclaim. We should 100% do the (or some other plan if there is one better) test us both plan today, because if we both IC ourselves then we take tomorrow from an assumed 2/7 to 2/5 odds of randomly hitting scum, which is a better than 10% increase. If one of us lied, D2 we most likely have a 1/5 scum to look for while verifying with certainty which one of the 2 drunk mayors is scum.
The fact that all of this is this case is proof enough that you're not scum, really. That is just not a position scum wants to put themselves into.
The fact of the matter is though, we will probably still want both of you on the exile wagon if we don't decide to go for some elaborate test, because it's going to be hard to get a town-controlled wagon otherwise. Thus we will most likely gather some information by the end of the Day even if we don't actively work towards it.