I wouldn't fret too much about shooting 22 LR in the Magnum cylinder from a safety standpoint. It is true that the LR case is almost 0.020" narrower because the Magnum has a bullet that is both about 0.003" wider and has no heel. By the same token, that makes the throat loose for an LR and the bullet jump long, so the pressure tends to be low when the LR is fired in the Magnum chamber and the main issue to concern yourself with is lead splatter building up in the mag chamber that needs to be cleaned for best accuracy and function with the longer round. Chambers for both cartridges go from case to bore with a taper rather than a step, that being a 30° half-angle for the .22 LR and a 15° half-angle for the .22 Magnum, so the loose-fitting .22 LR bullets in the Magnum chamber don't have a sharp edge to scrape against on their way into the freebore of the throat. Low pressure and inferior accuracy are to be expected from the .22 LR in the big chamber.
A couple of things can be looked at as to the cause of your sticking issue. When the gun fires, the case backs up. In most centerfire revolvers, the primer backs out on firing and then is reseated when the solid case head subsequently backs up into the breech face. After that, the empty can fall forward again and rattle loose in the chamber. In a rimfire, the case backs up and if the headspace is long, the sides of the case jsut forward of the rim can expand enough that it doesn't fall loosely into the cylinder again. That will cause it to drag on the recoil shield (the revolver's breech face).
You can measure the headspace to compare to rim thickness several ways, but the simplest thing to do is to chamber one round and fire it and, before cocking the gun, point it down and then hold it up to the sky or other light source and see if the head is still against the breech face. You could also try to slide a feeler gauge of one or two thousandths in between the case head and breech face if you have one narrow enough to slip in when the ammo has not yet been fired.
If that problem is happening, expect a caliper to show you the case is fatter just in front of the rim than in the middle of the case walls.
Another thing that can happen, if the gun has very rough chambers (I would not expect that from NAA) it is conceivable the case sticks to the chamber on firing and the middle of the head swells, causing the same jam-up. In that case, the side walls would be pretty uniform, but the back of the head would be domed. If you have a .22 LR rim thickness gauge, it would register the rim as being thicker if that happened.
If the chamber diameters are loose but the throat is not, you might get extra powder blown back between the case and chamber causing the brass to stick to the rear. That can cause it to drag on the recoil shield, too.
Finally, check the barrel/cylinder gap of both cylinders. Make sure pushing the cylinder forward does not cause the face of it to drag on the back end of the barrel. This causes scratch marks on the cylinder face you can see if it is happening. Having a case push back against the recoil shield may also push the cylinder forward (via the widened area in front of the rim) and if that causes the face of the cylinder to scrape against the back of the barrel, the drag just gets worse.
The solution to all the above is likely to be cause for getting the manufacturer involved. I always resent the cost of sending a gun to the factory, but sometimes it is worth it. Talk to the factory and describe what is happening. They will have seen it before.