The magnetic force depends on v which depends on the inertial frame of reference. Does then the magnetic force differ from inertial frame to frame? Is it reasonable that the net acceleration has a different value in different frames of reference?

Hint: Use the concept of relative motion.
Yes, the magnetic force differs from inertial frame to frame. The magnetic force is frame dependent.
The net acceleration which comes into existence out of this is, however, frame independent for inertial frames.