Gene Fullmer is usually considered a walk in bang up slugger, and true enough he used this style against Ray Robinson, but that wasn't his only way of fighting. Against another straight ahead slugger/volume puncher Carmine Basilio, he used his own awkwardly clever and effective "boxing" style of going in reverse and countering off the ropes. That's how he beat Carmine, and how he salvaged a draw against Dick Tiger. I think that's exactly how he would do against Jake LaMotta, and contrary to how probably everyone else here will see it, I think that Fullmer would decision Jake. It sure wouldn't be pretty and no one would think they were watching Willie Pep, or anything, but it would have worked for Gene. A heavyweight parallel to this fight would be the 1966 fight between Oscar Bonavena and George Chuvalo, where Bonavena surprisingly didn't meet George head on for a slugging match, but "boxed" his way to a close decision. Like Oscar, Gene Fullmer wasn't totally one dimensional as a fighter.
If Gene wades into LaMotta and looks to brawl with him, he loses plain and simple. Jake did mingle with lightheavys when already past his prime and held his own with them... and showed that he could bull the big boys around when in close. Gene while exceptionally strong himself, was not as physically strong as LaMotta, and he would find himself being bullied around on the inside with the Bronx Bull much the same way he was against Tiger. However, if he elects to box Jake then his chances of victory increase substantially, although it would still be a pick`em IMO. I think it will come down to what the judges favour, Jake`s aggression or Gene`s countering... tough call either way.