The Slaughterhouse cases in the 1870's essentially neutered the privleges and immunities clause(although the Chicago gun case is seeking incorporation of the 2A through the privleges and immunities clause). So using that's a no-go right now.
I think DLs are recognized by states out of convenience, I don't think the Feds forced states to recognize each others DLs.
Although I believe it is a constitutional right, it technically is not enforceable on the states, especially those with no RKBA amendment in their constitutions, like CA, or a very weak wording like IL, who also does not have a CC permit system at all.
Nationwide CC probably will happen after passing laws on a state-by-state basis, with all states eventually recognizing each others permits, barring a federal CC bill passing, which is not likely right now.