I don’t disagree with all of that, though it’s not what this OP is looking for. But I’ll point out that if you put higher burdens on gun owners wrt having them have to pay more it’s going to, from a practical perspective, skew gun ownership to wealthier citizens, or at least make it very difficult for poorer citizens to own a gun. Same goes for extra licensing. Often we are told that it’s too high of a burden for citizens to have to get voting ID after all. Gun ownership is a right, so you don’t want to make it so that only some citizens can participate, even though even now that is the case to a certain extent.
I will point out that we don’t put similar burdens on alcohol or alcohol consumption. Basically, if you are of age you can buy and use alcohol. This, despite the fact that more Americans are killed due to alcohol than firearms. Similarly, we don’t put those same burdens on tobacco use, again despite the fact that even second hand smoke deaths are more than firearms and nearly as many as alcohol abuse. And the later two aren’t protected rights.
That said, I have no problem, in broad terms, with licenses or background checks. Limiting firepower, depending on what you mean by that, is already something that has restrictions on it, as does the action (i.e. we already have restrictions on automatic verse semi-automatic or other actions with lower rates of fire). All of these things put the burden on the law abiding citizens, and, frankly, put harder burdens on the less wealthy citizens than the richer ones, which is, IMHO, a bad thing.
What about what the OP is asking for? What burdens could we place on actual criminals wrt gun crime? I think those I mentioned are the low hanging fruit that are fairly obvious. Not sure what else you could do. Basically, a person is a law abiding citizen until they aren’t, and when they cross that line we then prosecute them. Would putting harsher or more focused sentencing specifically on gun abuse do more? I think there are examples from alcohol that say yes, they would. Also, there is the civil aspect…if one commits a crime using a gun then perhaps open up more avenues to civil suits. Of course, again, we actually DO already have some sentencing that is geared towards gun abuse committing a crime, just like we do for alcohol abuse, but perhaps we could do more on that score as well.