You're not going to be able to reinforce that from the inside. It's inaccessible. Any benefit from reinforcing from the inside is negated by how much cutting you'd have to do to access the area and the incredibly poor job that would result since you still wouldn't be able to access it well enough to get a good layup.
Since you have to remove the gelcoat to repair it anyway, I would sand a bunch of inspection areas before I seriously considered a repair. Once the gelcoat is gone in several small patches, you can see the extent of the involvement of the glass. The hull is not actually opaque once the gelcoat is gone, so you may be able to actually access the under side of the tray through the battery box hole and shine a light through the under side.
If you want to reinforce it (I'm not saying that you should), you should do it from the outside. You should remove the gelcoat completely if you are adding glass. It's not going to take a beefy repair if it's already mostly structurally sound. It'd probably only take 1-2 layers of 1208, then once it's cured, skim it with epoxy mixed with microspheres, sand as smooth as something that's hidden under turf needs to be. You might need more glass if you find that it's heavily damaged, but based on your description, I wouldn't think so. 2 layers of 1208 wet layup is about 1/16" ballpark.
All that said, it's probably not an issue. Only concern is that you said it flexes some in those spots. That is a sign warrenting further inspection but not necessarily further repair. Maybe consider 9mm under padding to help distribute the landing.
