Can you breed them with other very similar cats in captivity to introduce genetic diversity, then keep breeding them with more Iberian lynxes so that they're mostly still Iberian lynxes?
Crossbreeding is necessary if you have too few specimens left to prevent inbreeding, but if you don’t, I think you’re better off using the entire population for a breeding program.
If you have sufficiently many “more Iberian lynxes” to get rid of the foreign DNA in the population after mixing that in, wouldn’t you also have sufficiently many for a breeding program that doesn’t introduce foreign DNA?
> Can you breed them with other very similar cats in captivity to introduce genetic diversity
Not.
Or at least not a thing to do without a lot of previous planning and backups. Lynx hybrids can carry a sterile gen.
Messing with biology is messing with millions of years of fine adjusted DNA technology. Would be like finding a marooned alien spaceship and starting to press buttons randomly. At this time we are just struggling to understand a small part of this natural technology.