The bridge you mention is the only road to Windermere and it is private. Windermere is a gated community where some folks take privacy fairly seriously. There's a guard/gate at the bridge blocking access unless your name has been left at the gate by a resident or you can come up with a more convincing reason to proceed than you want to explore some promising bonefish flats. There is some great fishing on the west side of Windermere. I've fished there out of a skiff. If you can meet someone on Windermere and get invited over, fishing should be good.
There are enough gas stations scattered around in the settlements that this shouldn't be something to be overly concerned with. Just try to not run low on petrol after 5:00 p.m. or so and be sure to fuel up on Saturday if you've got a long way to drive to where you want to go Sunday as most are closed then.
The Eleuthera Institute and the projects they are engaged in is very interesting. I was able to get a tour with one of the staff members. I'd suggest calling ahead of time to see when someone there might have time to show you around. Plenty of good places to fish out that direction too.
If you're going to spend time checking out the creeks around Deep Creek you might want to consider renting a kayak from Donald and Kristel Anderson who live there and run Eleuthera Tours. Some of this area has soft bottom and is better accessed via kayak than wading. Donald can help get you headed in the right direction for some good fishing. Their phone #: 242-225-4408 or 242-557-7309.
I'm going to be on the island during the time you are. Maybe our paths will cross. Have fun!