What's under the subfloor? If bathroom is above another room or crawlspace, be careful not to cut into the floor joists. You wouldn't want to create a weak spot right where all the weight of a full tub will eventually sit. Considering the relatively small size of the part you need to replace, I (very occasional DYIer) would definitely go with a dremel over a sawzall.

Hey! I had a similar problem when remodeling this original 1961 main bathroom in my house. The subfloor had water damage even worse than yours and it had been laid before the interior walls, so removing it would have meant demolish half the main floor and I wasn't ready to do that....So based on my very limited experience, you need to first make sure nothing's leaking and that the damage is actually and only from excessive moisture in the carpet. Then , what I did was dremel the whole perimeter of the piece to be replaced. Took a bit of patience and few blades, but it wasn't difficult. I then added a few pieces of wood where needed to properly screw and support a new piece of plywood in.