The good news is that I think you are correct that there is no structural damage. What I would do is remove the damaged gel coat to take a look underneath, just to verify. But, I think you're fine.
The first trick to this repair is doing it in the water without contaminating the water. Please do make every effort to catch your sanding mess and not just let it go in the water. Not using power tools will help, and frankly, this isn't really a power tool sort of job anyway.
The second trick is just matching the gel coat color. White is never white - it is white-ish. If you are OK with close-enough, then this is a really simple repair. The repaired spot will be whiter than the surrounding area. If you want the repair to be invisible, you will need to color match, which can be tricky.
You'll want to sand the damaged area to remove any loose material. You'll also want your sanded area to feather out over the healthy gel coat ... maybe about a quarter-inch to half-inch. Clean the damaged area and immediate surrounding area with acetone, wipe off, and repeat a couple of times. There could be wax, which you do not want under your repair.
If the damage is relatively thin ... let's say 1/4" ... you can just fill with new gel coat. If the damage is deeper but not structural, you might want to fill with filler to get the depth closer to 1/8" - 1/4", then when that is cured, sand and gel coat over that.
The key is to not leave loose material under your new material, to only apply new material onto roughened surfaces, and not to leave voids (air pockets) under your new material. If you follow that, you're nearly done.
Once everything is cured, just wet sand with increasingly fine sandpaper until your repair is blended smoothly. This is the part where environmental stewardship gets tricky. There is a good chance your marina will frown on you doing this at all. But, if you do it, you might try taping a "bib" around the bow with something like Gorilla tape and securing something like a milk jug under a hole in the bottom so that bib acts as a basin to funnel the wastewater into the jug. Wetsanding doesn't take much water - just a trickle - so you can be conservative.