The CBBT is crossed hundreds of times a day by RVs, campers and trucks of every stripe. So long as you meet the height restriction, you will be fine. It's one of those road trips that should be on everyone's bucket list. Yeah, it's not Highway 101 on the PCH or Grand Canyon's north rim, but it is quite the spectacular engineering marvel. The roads are divided for much of the journey, only going through the two 1-mile tunnels are you back on one-lane-in-each-direction.
Seagull Island, about seven or so miles up from Virginia Beach, is an OK stop, but can be problematic in a large RV - its entrance is off the southern end of the tunnel there, and makes for an interesting U-turn as you prepare to leave and head north again.
The best times to cross, IMO, are early in the AM and late in the evening, to see the sunrises and sunsets.