I would try and clean the mylar before trying to remove it. There are a lot of products out there that can do a good job cleaning it and you may get it to a clear enough state that it won't bother you.

Removing it would be a ton of work. You also may not be improving the game by removing it, especially if you muck it up while doing the work. I have found that when I take things apart and put them back together, it is really easy for me to over tighten a bolt, push to hard on a wire, crack a plastic, etc...

That is a hell of a lot of parts in buckets for a month. Hope you will be able to remember where the all go.

The hair dryer is the best method for decals, but you have to work slowly. The hair dryer is awesome because when you heat the glue, it comes up with the decal and you don't have to do any scraping.