I have a 1984 Handy Gun II 30-30/20 gauge combo. With the 20 gauge barrel on, there is no lateral movement, but with the 30-30 barrel there is a small amount and just like yours, it locks up tight. I tried shimming it with a BCS (beer can shim) just to see what would happen but then it wouldn't close all the way so the hinge pin isn't the problem. I believe it has to do with the sides of the barrel lug being slightly too narrow. Where it fits down into the reciever there is a small gap which is what causes the side-to-side wiggle. I measured the lugs of both barrels and the shotgun lug is slightly wider. So, any shims would have to be placed on the sides of the lug.
It shoots fine so I'm not going to mess with it.