Depending on their location, I'm sure those small impressions could be filled by TIG welding without overheating the putter head. You'd just need to find out what type of steel (probably some sort of 300 series stainless) you are working with then grind/sand/polish to match the surrounding finish.
I think it's possible. I have some prescription sunglasses I'll wear while golfing sometimes. I will sometimes have trouble with ball contact while wearing them. I think it's because my brain thinks the ball is in a slightly different place because of the vision correction. So if your hat is somehow impeding your vision, then maybe.
Off topic but ...
Coincidentally (or not), I got a shingles vaccine Friday after putting it off or forgetting for a good while. It is nothing to mess with, good luck. I learned of this disease from my little brother getting it. It was not easy and it never fully goes away (to my understanding). It is everywhere, non-discriminating it seems.
Hang in there; it gets better.