I think that's probable to be constitutional (even though I personally don't think so, but likely by virtue of the fact that other rights, like voting, can require registration -- this one I don't feel that I'm "Constitutionally scholarly" enough to comment on), but I don't know that such a measure would pass Congress.
There is a serious fear out of firearms rights folks that registration is a watershed event that leads to future confiscation should the winds of RKBA change in the future. Regardless of if it sounds like slippery-slope tinhattery, there are a good chunk of firearms owners (even the ones that the left considers 'legitimate' firearms owners: hunters and shotgun sportsmen) who quite seriously are scared of this.
In addition, there is the obvious cost of such a measure compared to what benefits in combating crimes committed with firearms. Certainly the failure of Canada's firearm registry to have any worthwhile impact despite its immense cost and effort can be pointed to as an argument against it.