Would replacing my screen jeopardise the watch’s waterproof properties

There's no way to guarantee that it would still have the same water resistant rating. Also, the replacement adhesive strips I have found so far are a bit thicker than the OEM one. So after you re-set the screen in place, it sticks up a teeny, tiny bit higher than it did originally. Literally, we're talking a fraction of a millimeter, but it is enough to feel. And if you really look closely, you can see the difference too. Does that matter? Maybe, maybe not, but personally, I worry a little bit about the edges of the screens being that much easier to chip.

Personally, before I did anything else, I would try picking up a good screen protector (probably one of the plastic ones like the stuff from Armorsuit, not a tempered glass one) and see if that covers up the scratch enough to not bother you so much.