The Republican Party has a chance, I think, to distance themselves from the Tea Party, while courting the Latino vote by actually supporting immigration reform--or at the very least, by not actively opposing it.
It wouldn't even be that hard--the President's doing the bulk of the hard, unpopular work for them, and if they wanted to, they could take this opportunity to put together a bill that would help them shape that executive order into something that would appeal to the centrists of their party that feel like the party has left them behind and become too extremest.
And make no mistake, establishment Republicans REALLY want immigration reform. Big business needs it, and so do Republicans candidates from areas with growing Latino populations. President Obama's taking care of the touchy-feely side of reform--limiting deportations of undocumented families and that sort of thing that Republicans don't want to deal with--they could be drafting something that plays to the hard-nosed enforcement and limitations that Democrats don't like to deal with. I'm sure that Obama would sign any bill that they sent him in the name of bipartisanship, so long as it's *some* token effort at reform.
But, you're right; and they probably don't do that. Establishment Republicans are afraid of the Tea Party, and the Tea Party won't accept anything less than mass-deportation. Republicans have a great opportunity that they can't take because if they do, they'll be destroyed in the primaries by a Tea Party candidate.