Unless your engagement is with the police, they are out of scope and you are not allowed to test them. If someone called the police, you already lost, actually. You should stop them right before they do that (friends of mine who do this kind of stuff work in England, where the emergency number is 999 and their principle is that they give up when someone dials the 2nd "9").
Specifically, you do not have permission to attack the police. Your slip doesn't cover lying to the police or presenting forged documents to the police.
When acting within the scope of your permission to attack, forged documents are typically fine, however I would definitely include a mention of such tactics in one of the documents signed, either the permission itself, or the offer or something else appropriate. Just in case anyone is not a fan, you want to have something written that says "but we agreed that such methods are ok to be used".
There have been a number of recent cases where pentesters got into trouble for exceeding the scope of their engagement. Don't do that. Always stay well within what was agreed, and if you ask yourself if a specific method is fine, that is a very good sign that it should be explicitly stated somewhere as a method you might employ. I mean, if it isn't clear to you, it probably isn't clear to the customer, either.