The Cruising Guide indicates the best place to anchor in Red Hook is in Muller Bay. Other than just the general busyness of the area, is there any reason I'd be ill-advised to do so on a Sunday afternoon and dinghy over to Fish Tails in order to pick up some passengers?

If you're lucky enough to
find space that would be fine and it's far enough in so you get very little ferry wake otherwise pull into American Yacht Harbor.

Will AYH allow me to tie up or are you suggesting I anchor in the vicinity?

They will charge you for drop off/picking up passengers.

You can pick up by dinghy aside of Fish Tails on a little sand beach. If you take on fuel and water I doubt they would charge for passengers at the fuel dock. You might call Island Yachts and see if they have an empty mooring available for a short pickup.
Anchoring out is really no problem short term.

We anchor out. AYH charges $35 for pickup/dropoff regardless of how much you spend on fuel/water/ice. But, they will let you hit the Neptune fishing gear store while you are on the dock.

We anchor out and dinghy over. It is a wet bouncy ride. It might be worth paying $35.

Cheers, RickG