Honestly, I don't think immigration policy is what causes the Latino community to support Democrats. I think that for the most part the Latin community has a more collective mindset that does not jive with the Republican ideology. When it comes to social issues they are more conservative (like us in the Black community) but this does not within itself mean they will switch to the GOP if immigration were no longer an issue.
It would behoove the GOP to find a way to repackage their message to convince the non-white, middle/low income people that their policies would work FOR them and not AGAINST them. Too often I hear conservatives here talk about those on the lower economic ladder as people wanting hand outs and not wanting to work...tell that to the abuela who workers her behind off for minimum wage but is on food stamps.
As a former Republican I actually think the GOP could re-frame themselves as the champion of the working-class (at least in front of the cameras). They actually almost pulled it off during the election... they were talking about the REAL unemployment numbers, bailouts of corporations, people not being able to find work, etc. If I were them I would increase the "class warfare" rhetoric and drop the social issues.