At the time of K-219's sinking, I don't believe the Soviet Union had any Deep Submergence Vehicles (DSVs) capable of going that deep, but they already had the Mir class of DSV in development in Finland, and they were delivered a year later in 1987. The Mir class are supposedly strictly civilian (maybe that's true, maybe not.
shrug) They also ordered two military versions of the Mir, the Konsul-class, but those weren't completed until after the fall of the Soviet Union (one in 1999, the other in 2009.)
Any of these could visit the wreck at least in principle, but recover the warheads? That probably depends on the condition of the wreck. If the warheads were thrown clear they might be easier to recover, but harder to find. If the warheads are buried in the wreck, they might be nearly impossible to recover without cutting up the wreck, for which you'd probably need more heavy duty equipment than these DSVs.
Somewhat related, the Kursk sank with nuclear weapons and was cut in half and partially raised to recover them. But that was a very shallow wreck, just 108 meters. The salvage company Russia hired for the job made a neat documentary about it: https://www.youtube.com/watch?v=uQJ6IMREvz8