No, flying with your non-dominant hand is not really an issue at all. In fact, most pilots learn to fly with their non-dominant hand, even in aircraft with traditional yokes. This is because most airplanes are designed to be flown from the left seat and most people are right-handed. When in the left seat, you fly with your left hand on the yoke and your right on the power (or other controls, as necessary.) When in the right seat, you do the opposite.
As far as the amount of force is concerned, for larger aircraft, the force you have to apply to the stick is created more or less artificially (or, at least, it's mechanically scaled down, such as with pneumatics/hydraulics.) In newer aircraft with electronic controls, the resistance force to the pilots' input is created entirely artificially. If the controls were linked directly to the yoke in a large/fast aircraft (think a jet airliner,) it would be physically impossible to manipulate them by hand, due to the forces being applied to the control surfaces by the wind. Small aircraft, however, do often have direct linkages from the controls to the control surfaces, but, even so, it's very normal to fly them with your non-dominant hand, even for pilots who have never flown before.