Now interestingly, building muscle for your problem probably isn't going to help and could potentially worsen the problem, though light exercising and cardio is always beneficial no matter your age.
if you've already invested in a decent comfortable strap and that hasn't alleviated and a lighter bass isn't an option (either though finances or choice) i would seriously consider looking at your posture and the position of your shoulders and elbow alignment when playing.
you could have a problem somewhere around your back which is resulting in referred pain to your shoulder.
The first thing i would do is seek a physio or sports therapist, and it might be worth taking your bass along too. A deep tissue massage and other therapies like acupuncture might help alleviate the pain and break down any scar tissue that might have built up causing a diminished mobility and extra pressure placed on your shoulder.
Your posture could also be a factor, in the interim i would suggest, if possible slinging your bass on to your good shoulder between songs and if theres any extra time between songs placing it on a stand, like when theres a guitar change, prolonged tune up or technical difficulty and if your singer has a tendency to talk alot between songs, take any opportunity you can.
Warming up your arms and stretching your upper body before you go on stage would be a good idea, and also id check out buying a high density foam roller to roll on at home (you can get them from amazon etc)