Make sure you have something to purify some water with. Its going to be hot and your going to need lots of water. You can pick up Portable Aqua at Wal-Mart for a reasonable price. There are other options, including filters, but since you don't go camping much its not going to be very cost effective for you.
Hammocks should be great this time of year. I actually just made my own a week or so ago and can't wait to try it out. If its supposed to rain, even the slightest chance, bring along a tarp to pitch over yourselves.
Make sure you have clean clothes to change into when you get to camp. You're going to be soaked from riding all day (as I'm sure you're already aware) and dry clothes will be great. Plus you don't want to go to bed in wet clothes if the temperature happens to dip.
Bring some kind of cover for yourselves to use in the hammocks. This could be as simple as a spare sheet from home. A sleeping bag would work too, just open it up and drape it over you like a quilt.
I'm not sure if ground fires are legal on the Berryman, although I'm sure someone here will be able to chime in and tell you for sure. On the chance that fires aren't allowed you'll need to bring some kind of stove if you want warm food. The MSR Pocket Rocket is a very light and popular model. If you want to get creative you can build your own alcohol stove. The Supercat
is an easy to build, cheap, and very effective design.
With just the basics of food, water, shelter, and dry clothes you'll make it through the night. If you want to bring along some extra things like books or a deck of cards it may make it more enjoyable.
Just remember that you're going to be carrying all of your gear on your bike, so more gear is going to effect your riding.
Have fun on the trail! You all are braver than I, I'd never attempt the Berryman on bike, there were too many places that weren't much fun on foot!