I never even thought of trying to get a screen protector, as an older camera there are few options but I have found a couple.
I wonder if I was to get instead a phone protector screen, and cut it to size. The screen I have is not touch screen so its just to protect it from my nose like you say and other things.
I was (and even after talking to specialist in shop) still am a bit unsure. Cameras, like bikes, get used, sometimes get bumped into things sometimes get a scratch, it's part of being used and wrapping in cotton wool ... they get used and get to show signs of use.
I always disable touch screen function of the rear screen as I find it a real nuisance. The screens are too small to do much beyond a brief glance at the image so a scratch or abrasion is not a big issue.
I still feel put efforts into protecting lens (front element).
That said, I did fit a new rear screen protector recently.
Ian