Been going to "my island" since I was a kid.
I have stayed everywhere from the golf course (slept on the grass) to the jail house (charges dropped) and every thing from the cheapest to most expensive.
If your camping there is plenty of info available.
The Banning House is very cool if your staying at the Isthmus.
If your staying in Avalon,
My vote goes to Hamilton Cove.
Your far enough away from the city to have some privacy and peace.
You have access to the private dock, a nice cove for fishing and snorkeling etc. and a bunch of other amenities.
Every unit also comes with your own golf cart so you can explore what you want when you want to.
I usually I bring my gear and rent a kayak for a couple days and strap it to the top of our golf cart. I launch from Hamilton Cove most of the time.
Here is one of the kayaks I rented and the Hamilton Cove Launch:
Here is the view from the balcony:
I would not ship my PA over there, my Trident? maybe. But doubt it.
I prefer to just rent what I need and not worry about shipping or dealing with the kayak storage and transport issues.
Good luck
I am already looking forward to our next trip over.