The way I see it: Larry just used his knowledge as a precaution in case Lee does something dangerous to him or his daughter. The whole reason for "hatred" is that Larry knew who Lee was before the apocalypse - a convicted killer, so naturally, he thought Lee was dangerous.
I don't think that Larry wanted to blackmail Lee - he didn't even tell Lilly about Lee's past while he had a chance to do so without any possible backfire. He just said that "there's something to this guy".
He was an asshole, but not a bad person - that's how I see him. He wasn't the smartest type, but he did his best to protect Lilly and himself. I don't think that his actions or decisions were wise and acceptable, but that's what he thought was best.
He was always pragmatic, I can even understand some of his reasoning behind throwing Duck out in the drugstore. Emotions and morals aside, they knew nothing about the zombies or about how it all works, so potentially, I would've also been cautious about a person cowered in blood that was in a close contact with zombies before coming to me. Kid or not, it potentially could be dangerous, it could harm the group or a person I care for. It's just that he presented this in a total jackass kind of way that made it look horrible.
In my playthrough, I tried to be nice to him, I tried to understand his intentions. He actually tried to help me with that zombie in a pickup truck, while he could easily stay aside and watch me die.
When Mark told Larry that I said that Larry just cares for Lilly and does everything to protect her, he was out of words. Which means that he couldn't really bring himself to say something really bad to me after hearing my real thoughts about him, which means that he wasn't a bad guy after all.
Of course, I tried to help him out in a meat locker. If things were different, I'm sure that my Lee could've changed Larry's attitude to "neutral" afterwards.
It is obvious that developers tried to make him look like a completely negative character, hence the red font and all the stuff, but he really wasn't so bad. For a "negative" character he surprisingly never harmed anyone other than verbally.
Now if we look at Ben, that's a complete opposite - he is not an asshole towards people, he doesn't hate you, yet he's responsible directly or indirectly for how many deaths in your group?
There you go. I chose Larry over Ben anytime, at least Larry can be useful for something other than whining