As long as you don't get any clumps I don' think it will hurt anything. I mulch-mow my overseeded areas too. You might also still have a few ungerminated seeds that would otherwise get vacuumed up. If you get clumps you can carefully rake them out or shoot them with a leaf blower.

Mow it. Don't do like I did a year or two ago. I let it get 5-6" high and a pretty good rain storm came through and dumped around 2-3". Next day I looked and most of my new seedlings were matted over and stuck to the ground where the rain flowed. I tried to rake them up to no avail. When that happens there's very little you can do to bring them back except to reseed all over again. Sooooo, mow it.

I have visited the Schulenberg Prairie at the Morton Arboretum, one of the oldest prairie restorations in the country. It is remarkable how devoid of weeds it is and I assume they only do occasional prescribed burns. If you do the early work, be it mowing or hand pulling or both, it should pay off in the long run. I use hedge clippers for the larger areas and hand pull the most obvious weeds. Whenever I spot ANY purple loosestrife coming up in the wetter spots I pull them by hand. I hand pull wild carrot and white sweet clover too. My one small prairie experiment is about 2 years old and I had annual foxtails pretty bad the first year. The grasses and forbs are so robust that I don't see much of those foxtails this year. Certain weeds, like the wild carrot, sweet clover, yarrow, and canada (among others) are adapted to a disturbed prairie environment and need some longer term control. Learn to identify smooth brome grass too because I have seen that invade really good quality prairies.

I think people will say to not mow if the seeds are starting to germinate. You could have mowed prior to that, but now it is probably too late.