Being conservative & pro-immigration is perfectly credible. It’s being conservative and pro-illegal-immigration that’s an impossibility.
Au contraire, my friend. The legal immigrants coming in are overwhelmingly candidates for welfare from the time they plant their first toe on our soil. If your image of the average immigrant is some scientist or inventor, you need to do some more research. And the legal immigrants are not voting conservative and are not apt to given the pool from which they come. We have two huge problems: (i) illegal immigration, and (ii) legal immigration. To think otherwise is to bury your head in the sand. Do some research at https://www.numbersusa.com/content/issues.html to find out what our disastrous current legal immigration policy is doing to us.