Ouch! ... but easily fixable ...
Securing the anchor & chain: 1st - why does your anchor move that much? Do you have a windlass? Even if you do, or even if not, but with a std bit or a sampson post, I put a UV-proof bungee from one point on the chain, around the deck bit, and then back up to the anchor.
Now my deck has one large Vetus Sampson post in the center of the rear of the pulpit. I take a piece of parachute cord about 18" long and it is tied to the chain link that passes next to the post. This gets tied around the s-post as a safety. This is really an "aw **** ..." cord for when I hit heavy seas while offshore ... it takes but seconds to un-do, and if I know I am staying inside and not heading offshore, I un-do it when I leave my mooring.
I also use this achor chain "stop" both as a safety and to hold the chain centered on the pulpit. Note that the pin can have some slop equal to the length of the ID of the chain link - that's why I use and recommend the bungee. If you had this, this should stop the chain from bouncing sideways and smacking your gelcoat. The bungee from this up to the anchor would keep it secure.
Repair: If really deep, I'd fill with Formula 27, I would not use epoxy as you do not get a chemical bond when gelcoating over epoxy. If needed, I can send you some cabosil (which is a thickener) for the gelcoat.