Hey all, found a user on Reddit talking about round screen protectors for watches but only gave a link to Amazon UK:
But for those in the USA, Amazon USA has them from a different manufacturer (KlearKare):
KlearKare Invisible Screen Shield Protector for Generic Watch 40mm Face Bezel | Military Grade Scratch Protection | Lifetime Warranty
A couple of things to look for is that this comes with an application solution. They don't specify what it is though they mention "NOTE: A small amount of water is used in installation." Folks may want to be sure to read the package and make sure there aren't any solvents in it besides water or very mild/dilute alcohols (I have been using Nikon lens cloths for cleaning DK2 lenses here and they have some isopropanol (rubbing alcohol) in them but it seems fairly diluted with water - which might be important for not hurting the lenses).
I haven't used these protectors so can't say yet how they are, but I'm going to order some. Price is $7.95 per pack of 2 with $1.01 shipping.
I'm sure there are others available as well but on the whole, these have pretty good reviews. Most are 5 stars but a few that rated it less than that seemed to have some kind of problem. Either the protector wasn't completely cut through (scissors would fix that) or they had some other issue.