I get the impression that UPS wants Heavy International experience very badly. Of everyone that I interviewed with I was the only one to get the offer and I was also the only one with heavy time. Most of my class has heavy international time.
So given those circumstances, I would focus on Large ACMI carriers like Gemini, Atlas, Focus, World and even EVA and Cathay.
Personally, I wouldn't take a job that I wouldn't enjoy, so don't take a terrible job just because it may lead to UPS-ANC.