Renting on the beach in La Jolla is an alternative to misssion bay. There is a boat launch there and kayak rentals right up the street.
Paddling through the waves is not for beginners, though, but once you get past the waves, you can go check out the caves, and the leopard sharks. It’s convenient, and the beach there is my favorite. Good idea to wear a helmet when paddling through the waves.
Mission Bay is much easier paddling but there aren’t too many easy places to both picnic and drop a kayak in the water.
I doubt you’ll be able to get your own ramada. I’d suggest bringing some easy-ups and tents and a grill to make your own space. You might consider dropping the kayaks in the water at a boat launch, then paddling them around to a nice picnic place.
Try Mariners’ point. Seems there is lots of grass and easy beach accesss, though it’s a long walk with a kayak from the parking lot – so drop in elsewhere and paddle to Mariners’ point where someone else has set up camp.
Be sure to paddle to the back side of Sea World and see the pengins, and I like to tie up on the dock at the barefoot bar, get drunk, then try to paddle back without falling off and drowning.