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?
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?