Actually, Begin's support for Argentina during the war is apparently OTL. Speaking as a Brit who's always been sympathetic to Israel, this seriously annoyed me when I first heard about it. I suppose with the POD in mind this could have been done openly instead of secretly, with the Israelis seeking to join in by harassing the British military bases on Cyprus or something. I've no idea how this would play out in the US though.
THAT would be a very interesting, messy POD to play with. Any ideas how the US would go if two of America's closest allies started duking it out over the Falklands?