True regarding the bird hunting, but you can always plug the gun. Almost everyone carrying a 5 shot gun is doing that. Plugging a long mag is no different than plugging a short one.
Go to the hardware store and just buy a wooden dowel - I like the thicker ones but make sure it's a little slimmer than a shell hull. If your mag holds 7, then measure out 5 3" shells worth of dowel and cut it (adjust this as needed - basically always measure out max capacity - 2 worth of shells). Open the tube, throw in the dowel, and screw the cap back on. The magazine should only accept 2 shells - if it accepts more or less than you got your length wrong. Make sure to test with both 2 3/4" and 3" shells as your gun should accept no more than 2 in the mag of either.
That gun won't be IDEAL for hunting but you could certainly make do with it.