This entirely rests upon the assumption that supporting amnesty would win Republicans support among Latinos, which simply isn't the case. All the polling of Latinos shows the primary reason for their inclination towards the Democrats is economics/welfare.
King made a good point about Bush's success among Latinos: it wasn't because he was some kind of economic leftist.
Republicans simply need to be at ease with the influx and increase of Latinos in this country, put forth Latino candidates, and chill some of their class rhetoric. This involves clamping down on the more unsavory elements in the GOP whose immigration stances are a thin veil for their racism. Bush did well with Latinos because he was from Texas, had Hispanic family ties, and seemed incredibly comfortable around them. His Spanish wasn't great, but at least he cared enough to try and didn't seem like a phony when he did it. Immigrants and their children are naturally drawn to an entrepreneurial message, which is something the GOP can tap into.
This isn't like the black vote, as much as Democrats are hoping it is. Republicans can certainly get 40% of the Hispanic vote if they put forth the effort and stop appealing to the worst elements in the party.
I live in a heavily Latino community (>50% in 2010). My observations of 2004 in my area are in agreement with UI's analysis.
