I wouldn't worry too much about the no-limit issue. CA's governor just signed a law banning ALL lead ammo for ALL hunting starting July, 2019. I can't imagine there will be many guys who want to burn expensive, non-tox shells on doves. The populations will explode and then there may be too many of them.

From AB711:
"Requires use of nonlead ammunition for the taking of all wildlife in California, including game mammals, game birds,
nongame birds, and nongame mammals, with any firearm."

Dove hunting in CA is pretty much doomed, unless somebody mounts a successful challenge to the law. So CA will probably have enough Eurasians to supply the rest of the country.