The main obstacle seems to be the Bush admin's fear of a refugee movement bringing stories of the horrors of the war in Iraq. Perhaps also some exagerrated fears of terrorists infiltrating among immigrants, which some candidates claimed could happen on the Mexico border during periodic bouts of immigrant-bashing.
A simple enough POD would just involved Bush changing his mind, making a public case that taking in refugees was part of the US's duty as a beacon of freedom. Bush, for all his huge faults, has never been anti immigrant.
I'm also curious if there's any POD which would've brought over large numbers of war bridesmarried to GIs, much like in the past from Vietnam, Korea Japan, Germany, or England.