I have arthritis in my hands and cannot tolerate weight on them at all - nor can I tolerate road bars or road brake levers. This is more due to the position that road bars and levers puts my hands in than anything else.
I've trigger shifters and mountain bike brake levers on two of my bikes, and SRAM X0 grip shifters and mountain bike brake levers on my road-go-to-bike. Combined with a very upright riding position + short reach it works well for me, I can literally ride all day long without pain.
It sounds like your arthritis is worse than mine, and a coaster brake could be your best solution, but don't give up in brake levers yet. A qualified professional fitter who knows his or her business might be able to develop another solution for you. Not all fitters have the same skills or are accustomed to working with more than minor physical limitations so if there is more than one LBS in your area ask around to find out which one has the best fitter. That being said, a good fitter is expensive.