Using your definition of "lib" as someone who favors big government vs. individual liberties, there is no connection to liberal philosophy and suborning illegal immigration.
Moreover, as Top Gun points out, and as has been pointed out many times on this forum, the Republicans had six years to fix immigration, and what did we get? zip, zero, nada, that's what.
Some things just aren't "conservative" vs. "liberal" issues
Both parties would like to obtain that demographic at the polls, no question.
But would the Bush DOJ have sued AZ ? Won't ever know of course but I have trouble thinking they would. Given that they did take public sentiment into account (when pressed ala the Bush/Kennedy Amnesty bill)), seems unlikely.