1-Shot Cop and Full Cop in a closed setup just makes very little sense
man, I'm not even sure I buy that you're a cop. Even if you're town, I totally see you claiming cop to get your targets lynched. You even said day 1 "what if I claimed day cop?" so it doesn't seem far fetched to just fakeclaim daycop. say you're right, and P is scum. If we lynch him and he flips, noone is going to suspect you, even if you come clean with your fake claim.
That'd be pretty bold, even for me.
I actually want to come back to this. If I happened to be 80% sure (which is less than what you claimed) that Player X is scum, and if claiming cop is believable, I would do it. It's not even a hard choice. We get him lynched, and I admit that I was fake claiming. If it worked, noone is going to have a problem with that. If it didn't work, the worst case is that we lynch 2 towns. I'd trade an 80% chance for a scum lynch for a 20% chance of 2 mislynches any day, any time. Especially in the position we are in.
You said, and I quote, you are so confident in the P lynch that you are okay
if we lynch you first and then lynch him. This is in every single way a worse scenario than the one I described above. There is no good reason not to fake claim cop.