It is a pretty urban area. The old guidebook lists George Washington Campground 30 minutes away in Chepatchet, RI (on rt 44) as the closest. I have no idea what it is like. Alternatively you might find a cheap hotel around Attleboro. Unfortunately it is not really a camp and climb area.
Plenty of places off Rt 146 you can stealth camp if you just looking to sleep and cook some food on a stove. If you're looking for full-on campground with showers, etc, there are options near the MA/RI border. Other option is to go south towards Naragansett and the beaches, there are lots of nice places to camp down there.
You can spend the day at Lincoln Woods, and at night sleep at the rest area on I-295 it's call Rhode Island welcome center, which is also a bike pad, I do it other with my rv, in fact, you can get from there on your bike to Lincoln Woods on the bike pad and live your vehicle ay the rest area.