No, that's not how Amendment XII works. Under the scenario you lay out here, had Cheney remained an "inhabitant" of Texas, along with Bush, then Bush would have still ended up with the 271 EVs he did receive (following the brouhaha in Florida, that is), while Cheney would have ended up with only 239 (i.e., the 32 Texas Electors could not have voted for both Bush and Cheney, and, presumably, they would have cast their votes for Bush). Lieberman would have still ended up with 266 EVs.
But even though Lieberman would have had more EVs than Cheney, Lieberman still would have fallen short of the 270 EVs he would have needed to be elected Vice President. In that case, the selection of the Vice President would have devolved to the U.S. Senate, which would have then voted for which of the two men would be chosen. Per Amendment XII, the Senate would have been restricted to voting for either Cheney or Lieberman.
Funny thing is, though, following the 2000 election, when Congress convened on January 3, 2001, the Senate was split 50-50. And with no Vice President having yet been elected, there would have been nobody to break the tie had the vote ended 50-50! That would have been an entertaining scenario to see play out.
“when Congress convened on January 3, 2001, the Senate was split 50-50. And with no Vice President having yet been elected, there would have been nobody to break the tie had the vote ended 50-50!”
But wasn’t Al Gore still technically VP at that point and wouldn’t he have been in a position to cast the tie-breaking vote, possibly for Joe Lieberman? In hindsight, a Bush-Lieberman administration might have been an improvement over what we got.