What we are going on is Hagler's assets. Chin, punching. Don't get too hung up on this, nobody here is insisting Hagler would go to war with Lausse, but his assets are undeniable.
I've always thought Lausse was very underrated among history's MW contenders. Not only could he obviously punch, but he was very strong, and surprisingly shifty and skilled. Ironically, the win over Fullmer wasn't really considered one of his bigger ones at the time, but looks very impressive in retrospect. I do believe his next couple fights afterward, against Savage and Boyd, revealed some of his limitations. He tended to loop his punches widely and leave himself open, and hunt the head with left hooks. It's a shame he couldn't correct those flaws, because his game was really pretty impressive otherwise.
Tiger for sure. Between his iron chin and tight defense, he'd be the guy to get inside those wide swings and work over Lausse on the inside. Aside from that, possibly Zale, Cerdan, or LaMotta. How about Apostoli or Garcia? As others have said, I don't see Hagler doing it. I think he'd have to box and move at least for some length of time, like he did against Sibson and Briscoe.