Tbh I don't think Alastor hates Lucifer, I think he was so on sight with Lucifer at first because of a combination of jealousy and being generally annoyed with Lucifer's presence, and part of it was because he wanted to stay close to Charlie (his reasoning for that is up to interpretation, but personally I think there were multiple reasons he wanted to stay close to Charlie). The first part of Dad Beat Dad is just Alastor messing with Lucifer, because that's what Alastor does. He does it to Vaggie, he does it to Husk, he does it to Sir Pentious. He just likes to fuck with people.
I'm actually wondering if "fucking with people slightly sadistically" is a love language of his, because Rosie's way of showing affection towards him is similar. Telling him "You could learn a thing or two!" when she comments on how polite Charlie is (essentially calling him impolite to his face), pretty much telling him (again, to his face) that his manners are shit, she just generally teases the shit out of him and while her jabs at Alastor are more lighthearted than Alastor's jabs at literally everyone, she's still making jabs at his character. Alastor reserves his extreme sadism for rivals and enemies (Vox, for example) but if he just wants to fuck with someone? He does not hesitate to be a little sadistic.
I think with Lucifer, Alastor started off more like "this'll be fun, I can make jabs at his height" and then Lucifer started making jabs back at him (making fun of the Hazbin Hotel's name, most notably, which probably made things more personal to Alastor) so Alastor got more aggressive, which eventually culminated in Alastor taking the piss out of Lucifer wanting a relationship with Charlie. If you look at the background during More Than Anything, he doesn't look like he's mad about Lucifer's existence.
This doesn't look like a guy who's irritated by someone else's existence.
My personal theory for this is that Alastor was initially trying to drive Lucifer away to stay close to Charlie, but at this point he realized he didn't have to worry about it, which, by extension, means he doesn't have to worry about Lucifer. Dude has other priorities and is probably just as concerned about Lucifer as almost every other Sinner (that is, not concerned at all), he has no reason to hold a grudge against him except pettiness (which is something Lucifer does, if his response to Alastor showing up at the end of Finale is anything to go by). Alastor doesn't see Lucifer as a threat in any capacity at this point, so their rivalry is one-sided on Lucifer's end.
I'm excited to see where Alastor and Lucifer's dynamic goes from here. Lucifer has the pettiness of a fucking middle schooler and Alastor could not give less of a shit (which actually seems to be the case with all of his rivalries; you could argue he cared about what Vox was saying about him in Stayed Gone but counterpoint, he'd just found an excuse to spill possibly decades worth of tea and increase his listener base on his radio show, and fuck with Vox), and I think it'll make their dynamic really funny.
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