Bottlenecking the GPU will not harm the GPU in anyway. In fact it could "extend" its lifespan. If the GPU usage doesn't get to 100% due to be bottlenecked, it won't get as hot which in turn "lengthen" it's lifespan. I don't think it would lengthen to any large extent, but it certainly won't harm it.
As to whether the 8350 would be a decent upgrade, it is faster than the 6300 (in clockspeed) and has more cores. So it will reduce / possibly eliminate the bottleneck. It's not going to be as good as an Intel platform, but it will be a cheaper upgrade option. One thing to consider for the upgrade to a 8350 is your PSU and the motherboard. Is your PSU going to be able to handle the power demands of this system? Next what motherboard do you have? Many price conscience AMD motherboards have very weak VRM's with poor quality / no cooling for them. In these circumstances, the system could throttle the CPU to protect the VRM's or become unstable. So before making that upgrade, make sure you have a quality AMD motherboard. If you find that you need to replace the motherboard to accommodate a 8350, then you might want to consider switching to an Intel platform.