Can't say I've ever heard of a national forest actual designated campground allowing a generator to be running all night long not only in Michigan but in most other states or every state. That's what an inverter and a battery bank is for - the QUIET hours of about 8 or 10 pm until about 8 am. breathing assitance etc machines run fine on an inverter as 2 of our friends use them on their inverters every night when boondocking with no electric hookups. You sure don't need A/C at night in Michigan hardly ever as it's usually 60 degrees F in the summer at night.
Why would someone want to run a generator all night anyway? Might get away with it IF you are the only camper in the entire park and no ranger or DNR officer could hear it running. However, if they did hear it, you'd get a loud knock on your RV door and maybe even a personal coupon issued to you that you can redeem at the courthouse.
State parks never allow generators running at night in the Midwest or any other state we've been in and it's stated so right in the park rules you sign that you must agree to. Kind of a strange request but the OP must have a reason to be asking about it. There's a couyple thousand CG's in Michigan and we've been RV here and in the 49 of the 50 states minus Hawaii for over 50 years now plus a lot in Canada.