Since lots of the tags have talked quite well about love vs control (he's after the latter, maybe he thinks he feels the former but it's a moot point), and since this is the vampire story full of fucked up vampire ships, I want to look at sort of an unspoken element of this question: could Lucy have had a fulfilling relationship with Max?
With David, there's no question that his influence makes his partners (the boys, Star, Michael) worse. He's bringing them into a cult/gang (depending on if we're talking movie or musical), they ultimately answer to him, he is forcing them to do murders (even if just so they'll stay alive). At the same time, what David's offering does have freedom of a sort: freedom to walk away from society, freedom to hurt those who would hurt you, freedom of nontraditional relationship structures, freedom from disease at the height of the AIDS crisis. Even with all the costs, you can still see how Michael and Star could find a kind of fulfillment in vampirism.
I don't think that's the case with Max.
Movie!Max, no question to me, does not see Lucy as a person. He's too callous, confident he can overwhelm her into doing what he wants while brushing aside her complaints. Musical!Max, for instance, is doing his "haha unless" boundary pushing when he apologizes for snapping at dinner, but he does apologize--because he wants to keep Lucy sweet on him and trusting.
Movie!Max, meanwhile, responds to Thorn attempting to maul Lucy by inviting her over, saying "Thorn promises to behave himself this time." And then he kisses her despite her having the MOST uncomfortable body language in the world. Her fear is not important, her loss of trust isn't even important! She trespassed, Thorn did his job, now it's time to move on to the next stage of the plan. There's ordering for her in the cut scene as well, but honestly the kiss is always what sticks with me (and part of the difference is necessary in pure mechanics as well--movie!Lucy is so passive that it's honestly frustrating, and Weist isn't really playing it as trauma so much as she's just kinda....flighty? While musical!Lucy is skittish because of her ex but wants to be comforted, and needs to be reeled in with deliberation).
Movie!Max is clearly angling for the camouflage Lucy can give him. He seems to enjoy that she gives him a ticket to engage in the social rituals of a man in romantic pursuit, and how it gives him control over her, but at the end of the day she's Charlotte Haze.
It is, I think, EXTREMELY significant that the boys are directly under Max's thumb in the movie; he approaches Lucy specifically in view of the boys mooching around the shop, their only "rebellion" is trolling him with the kite (that scene kind of bugs me insofar as it "cheats"--you cannot watch it back and see ambiguity in Max's role like you can consistently with Nolan, they have Hermann play it pretty straightforwardly as Normal Guy, but it's still tame as hell in terms of what the boys are doing), and after seeing David's body Hermann's performance takes on a very "oh this guy has SNAPPED" quality to me. Sort of faintly talking about his plans, David and his boys (who are all quite dead), and then finally latching onto Lucy as something he can still have control over. I think movie!David has the trappings of a rebel and zero actual agency, which is sad and certainly what would have become of Lucy too.
Musical!Max is an odd thing. In some ways I think movie!Max is more human than him, in that he has a set of drives he follows and earnestly seems to buy into even when they are inconvenient to his goals. Musical!Max, meanwhile, is sort of the essence of a calculating predator---his persona can be both full-on and very mercurial depending on what he wants his target to do. He does have things he sincerely enjoys having in his life i think--order, stability, movies, someone impressionable in his orbit that he can mold--but he is powerfully calculating and long since remote from any true depth of feeling. (Probably, given this is Cycles the Musical, David would start to look a bit like him if he started getting toward quadruple digits).
I think he is earnestly attracted to Lucy, but I also think all living things are tools to him. He might not kill often, but he's replaced it was something weirder and more cruel: trying to exercise complete control over someone's will without them noticing. His interest in Lucy's "wildness" aligns with his own monstrosity, and if he brings bloodlust out in her he can indulge it while feeling like his hands are clean (and also make her feel more beholden to him). He is, perhaps, charmed by the idea of a Hollywood romance, and doing all the grand gestures, but it's a structure with parts, not a connection between people. There is no room in that for Lucy to assert her personhood--when she tries, Max's response is to back off but push something else subtly, until she's where he wants her anyway.
Likewise, musical!Max's vibe is DIVORCED. I think it's significant that they keep quite a bit of physical distance between David and Max (not even on the same levels of the stage in their one scene together!). There's a profound bitterness there; unlike in the movie, I think musical!David and Max were probably fairly close at one time, and Max is trying to get that dynamic of mentorship back however he can. He has Pete, then Lucy walks into his life (scholars may debate forever if he threw Pete to the wolves on purpose or just was too distracted by his shiny new toy to life a finger). Lucy can fulfill not only someone to be mentored, but also be an acceptable romantic partner. And Lucy is, I think, a gateway to Sam: a nervous 14-year-old nerd actively yearning to be back under the stability of living with his dad even though part of him knows it was an abusive situation and they ARE better off now. That situation has got to be like TURBOcoke for Max (honestly I think he's written Michael off from the jump, something to maybe bring David closer again at best, another hostage to keep David in line at worst). Pete 2.0 except this time you got him legally locked to you.
And for Lucy...she wants security, yes, but I think any allure in Max is artificial and likely to vanish as soon as his hold is comfortable. They don't have the same taste, same political opinions, same child-rearing beliefs. They have unbelievable chemistry, but that spark fades and eternity is a long time. I don't see him making more than token concessions to her and I don't see her giving over command of her sons to him without it absolutely destroying her sense of self, because she's handed them back to the very same suffocating hand she promised to protect them from.
