The 2 places I've been that have LAM residencies did not mention this preference at all, and at one it was even hinted that you'd have a better shot coming straight from vet school (or if you did work, worked in a lab animal position, or somehow demonstrated that you actually are interested/committed in the field as opposed to just throwing in an application because it sounds interesting - apparently this happens now that they're in the match program). That's why I find it interesting that you posted it.
I'm doing an ASLAP summer fellowship right now and I will ask them what their thoughts on the matter are. I for sure would not have assumed it was de facto requirement, at this point I'm going to assume it's very program specific.
I also find it weird because I don't really think GP's are necessarily good generalists, the places I've been it's like you see the same things over and over and so you get very good at doing those things.
PS - Just so you know they do mock ACLAM exams all over the US