I'm lucky that my in-laws have a little house in Southwest Harbor, which is on Mount Desert Island (MDI) kind of near the Bass Harbor light. I always bring my road or tri bike when we head up there from the Boston area. Over the years a lot of the roads in and around the park have been repaved, so the road quality is pretty good. After not having mountain bikes for many years, we just got new ones for myself, my wife and my 1st grade son. We plan to take both road and mountain bikes when we head up some time this summer. There are also the carriage roads that are awesome for riding - they are gravel, so would be ok for a hybrid, gravel, or mountain bike. My personal favorite is the park loop road, including going up Mt. Cadillac. I've done several 5-6 hour rides up there with 10k plus of climbing on the day, including a couple times up Cadillac.
So, I guess if I was in your position I'd stay in the Acadia area - that is what I'm familiar with. There is a lot that the area has to offer for biking. I'm going to be looking into more mountain biking options for when we are up there some time in July or August.
In terms of where to stay, if I didn't have a place to stay I'd look into renting a house rather than staying in a hotel. You can bring a lot of your own food along, which I think is currently the mandate in Maine. Most places on MDI would work fine for a location, as you can drive from one side of the island to the other in 30-40 minutes if you'd want to. We often go up in the winter, and we usually park near Eagle Lake for xc skiing, running, or biking, depending on the season. We love the Asticou Inn in Northeast Harbor for popovers, but I'm not sure if they'd be serving. The Jordan Pond House is also a favorite spot, but I don't know how they'd be handling the current situation.
One thing that is a bit harder currently is filling water bottles on rides or runs. This can be solved by doing loops to a vehicle or where you stay. I'd often fill up at the Jordan Pond House or the top of Cadillac, but I'm assuming water fountains are not on.
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