Watch videos by this guy. https://www.youtube.com/watch?v=lrrM4iVPVX4
Reverb has good advice for the fin. I’d go with a new box and removable fin.
I can’t stress how IMPORTANT it is to use a fin box template if routing the old box out. That would be the way I would go. If you don’t have one available it could get expensive to outfit for a one-off. You can make a very simple jig out of 1/2" or 3/4" plywood with a slot cut in it the same dimension as the fin box. Use with a top bearing 1" flute straight bit, preferably 1/2" shank but not vital. DO NOT DO THIS WITHOUT A GUIDE! Even with a Dremel, tape a straight edge down on the line to let the bit/collet/shank ride on to make a straight line. At very high RPM’s it can get away from you easily and very quickly to create a very ugly gouge. This board looks nice and worth the effort to bring back. The color tint is a boost in doing that to. This could turn into a beautiful restoration. Best of Luck.
For the fin box, I would use a router because that plastic will be tough. I would try to make the cut as tight as possible to avoid having to pour in a lot of filler.
Be very careful and use protection because that plastic or glass forming the old box may come off in larger pieces. You don’t want a piece of hard plastic hitting your eye. I don’t like cutting out old boxes, they always break when I do it. If you can get a helping hand, use a shop vac and keep the nozzle close to the router.
I use a smaller laminate router for most of my boards. I made jigs that fit the size of my router base for the 10.5" center box. I’ll cut a little, stop to make the cut deeper then cut more, and repeat because it is not a plunge router. For this kind of job a full size router would be easier. Dremel doesn’t have a long enough cutter to use a jig.
Router jig is as simple as a piece of wood to get the edge straight. A right angle will help you stop at the end of the box. Do one side and flip it around to do the other side.
I think there are lots of videos online that show how to do a basic fin box installation. Once you get the hole cut, it should be easy.