To make it easier for others, I downloaded it and attached it here.
I think you are going to need access behind it. What is there?
I'd be looking to install a more standard 4" Marine/RV type fixture.
The easiest thing to do might be to get a thin piece of starboard, cut it to over those holes and sand or route the face edges, drill it for the new faucet, seal it to the enclosure, and then drill through the pre-drilled holes through the enclosure to complete the install, which will likely require accessing the plumbing behind the enclosure.
You could also treat fixing the holes like any other fiberglass repair. I've twice repaired major defects in boat shower enclosures, both more significant than those holes, with fiberglass and epoxy, then thickened epoxy for fairing, and then light sanded and primed and painted the whole enclosure with Awl Grip. The repaired areas looked absolutely brand new.