> I’d like to ask for about 5k more than the max which would match my current salary in the private sector.

I assume this means that with a certain grade, the pay for the max step is $5,000 below your current salary. Whether you can get the max step is office dependent but there is no mechanism to go above the max step except for maybe a hiring bonus (and if the advertisement doesn’t mention a hiring bonus, there isn’t one). You can provide pay stubs to ask for the max bonus, a couple should suffice, and it depends on office if they consider things like bonus or commission (no idea if that applies to you). If they counter and say “this is the best we got” (whether that changes or not), that’s what they can do, and since you’re mid-level, they’ll probably just go to the next candidate after that.