Perhaps the best approach to universal background checks is

[Glaug-Eldare]

9. How about this?

I was thinkin' about this very question a while back, and I think I've got a way to get it done with a much weaker Constitutional issue, if any. Currently, only FFLs are required by federal law to conduct background checks, and only some states have laws in place to cover private sales. That could be made universal by tweaking the current law regarding FFLs, making a state's access to NICS contingent on creating a scheme to cover private sales. That is, unless a state requires and facilitates NICS checks for private sales, FFLs in that state will be completely unable to import stock, transfer firearms, or basically do anything. There should be minimum standards for access and affordability (e.g., it would be illegal to require sellers to travel to a remote police outpost in the desert and pay a $300 fee in order to sell a gun) which the states must maintain, or else face other penalties.

I favor a scheme like Maryland's, where if you choose not to pay an FFL transfer fee of $75 or more, you can have any state police barracks run the NICS check for you for $10.