I suppose it depends on your carry method, or case. You don’t need a proper case per say (for example I use a small soft guitar case that doesn’t even fit the whole shamisen, haha), just something that will make it less of a chore to lug around.
Once you have that I’d probably work on first securing the shamisen to the case, be it with stuffing, straps, tied string or parachord, just something to minimize things from jostling around too much in the case.
After you secure it to its casing, find some way to secure the case to yourself (this where a soft guitar design comes in handy). The reason I would opt for carrying it on your back is because that way you can feel the motion of the case better, and adjust your riding style to get the best pace while avoiding too much shaking around. Your body should also act as a kind of shock absorber to avoid that same jostling, and of course having decent shocks on your bike would help too.
It’s risky business depending on your location, but its far from impossible.
Overall, just treat your instrument like a small baby-child. Take things slow, and always be mindful of the shamisen, and your surroundings. I actually wiped out really bad and fractured my radius the first day of my summer vacation cause I was going to a relaxing outdoor day. You can imagine how relieved I was have decided against bringing it out with me that day.