THEORY: Asgore doesn't want Toriel back, he just wants the family together for the few days Asriel is visiting(and also to prove that he wasn't responsible for the bad thing that people thought he did during the Dess incident).
Toriel and Asgore divorced while Asriel left for college, who is completely unaware it happened.
Asgore considers how fucked up it would be if Asriel returns home to his dad living somewhere else, that it's better if short-term they just pretend they're still married and break the news once both kids have moved out of the house.
Toriel misunderstands and thinks Asgore wants to get back together with her, because Asgore's autistic and can't communicate for shit.
>but what about the chapter 5 hints? what about the jealousy? the black shard?
Asgore's a RED HERRING. He FOUND the black shard in Carol's house. Now think about Carol, she's a busy woman who doesn't have time to see her hospitalized husband, meanwhile Asgore has gotten closer with him, she must certainly know about this which would cause jealousy. Asgore isn't jealous of anyone, in reality he's fine with letting Toriel move on but he really wants to keep the family together for the kids' sake.
PREDICTION: In chapter 5, Carol will see something happening between Asgore and Rudy that will enrage her, she'll set the flower shop on fire. However, Kris, Susie and Noelle followed her and witnessed her crimes, they try to stop her and she opens a dark fountain trapping them all in so the shop keeps burning.