I can’t provide quarter sheet recommendations since my non-keyhole sheet came custom from a tack shop that is now closed, but yes, you can ride with a saddle pad under a full quarter sheet.
I prefer a saddle pad under since a saddle pad is easier to clean (especially if the quarter sheet is wool); I just choose something thin and pay close attention when tacking up to feel for the saddle pad’s placement, since you can’t see it very well.