I'm not sure if they will let you in to drive around. Easiest way to see the campground would probably be to take the boat from the Magic Kingdom dock. Then once you get there, walk past Pioneer Hall to the bus stop and ride the bus around. Would probably take you 20 minutes to ride through the entire campground. And a few of the loops are so close to Pioneer Hall that you can actually walk through them from there.
We weren't all comfy in the TT, either. Ours slept 5, so we made a pallet in the LR floor for the two boys. It was so crowded that everyone had to get up for anyone to be able to walk around. But we were all willing to put up with it just to be in FW. We'd go down for 2 weeks, buy annual passports, then the next year right before the passports would expire we'd go down for 2 more weeks.
Before we left home I cooked and froze most of our meals. We'd have breakfast, go to a park as early as possible, stay there until 1-2, then come back to the campground for lunch. Kids would nap, then after an easy dinner we'd make up all the beds and the pallet, then go back to a park until it closed.
The only problem was the bathroom. None of us wanted to use the bath house, so we had to assign shower times to everyone and scatter them out through the day. But it all worked out, we had numerous trips to FW, and we all have many fond memories of our trips together.
If you tow the TT down, you should probably plan on two days down and two back, especially with the kids. It's over 650 from here, and even with switching off drivers it's tough to make it in one day. If you're interested, I can give you some free places to stop on the way. In December we stopped at Flying J's, gassed up, and spent the nights there. Felt perfectly safe, lots of RV'ers there, and I'd do it again in a heartbeat.
BTW, where do you camp at the beach? We go to Ocean Lakes several times a year.