Cross stitch and embroidery

That’s no issue at all! Look at this:

You can clearly see loop marks but when finished you don’t notice it at all! This is even without iron / wash.

I’ve worked with small loops on huge patterns without issues.

You've already had some good advice there but one extra trick is to cut up some cheap felt and put it in between your stitches and the hoop/frame. Just helps protect things a bit more, especially if it's a wooden hoop that might splinter and catch on your stitches.

Sometimes doing what you're about to do can sort of flatten the stitches down more where the hoop was, but they tend to fluff back up at the end when you wash the finished piece.

(and yes I'm taking like everyone always washes their finished stitch but don't worry if you don't, I often don't bother either, only if it seems needed!)

You'd want to try it on something you don't care about to be sure, but I think it is something people do. I've done it on practice pieces before, but since I didn't care about the outcome, I don't remember how well it went. Probably if it was terrible I would remember, but that's not very helpful, lol.

It would depend on exactly what kind of stitches you were clamping, though. Something flat done in a relatively fine gauge is going to do a lot better than something with surface action, like French knots, and I wouldn't do it at all on beadwork. You'll also want to be careful how tight you pull the area, and how tight you make it, but I'm sure you don't need me to tell you that, lol.

Personally, I would use your situation as an opportunity to buy a bigger hoop or frame, but if that's not an option, I would try clamping on existing work. I mean, I have, so...