[ Before anything else, Bakugou would probably feel that it is extremely important to point out that he waited a long ass time before doing or saying anything about the weird trainwreck that the finals of the sports festival turned into. He stewed upon it. He thought and pondered extensively. For weeks. And for him, that is actually quite a show of commitment, and also indicates that he actually thought to himself that his kneejerk reaction might not be the most appropriate reaction. Character development is beautiful, basically, and okay — so maybe he thought it would make the whole "we're dating but we don't talk about it explicitly, ever, because honestly what the fuck" situation a bit awkward and reduce the amount of making out they do. So he has ulterior motives. Whatever, character development is still beautiful.
After weeks, though, he concludes that his kneejerk reaction was almost definitely the most appropriate reaction. His pride? It was offended, horrendously so, and it demands for justice. He wants vengeance, and also a proper victory, if he's going to keep the stupid medal and all.
He also doesn't want to cause property damage that he'd probably be held liable for if they went full-on fighting with their quirks in the streets, because (1) he is not rich, and (2) that seems like the kind of stuff that goes on your criminal record, which wouldn't look good for his future budding career. (Not that his future budding career is looking great in the first place, when his claims to fame include getting attacked by a sludge monster and then getting muzzled on TV, and hopefully too children weren't watching that because how would an appalled parent explain that, really?) As such, he's forced to wrack his brains over how he can get a rematch that appeases his ego without becoming a public nuisance.
Tl;dr: this was a disaster before it even began. But what else is new.
But the shorter recap of how they got to where they are goes something like Bakugou going full-on grade school bully and making sure that when he approached Todoroki with this brilliant plan, it was in front of the entire class. There was some weird egotistical and mildly Freudian grandstanding, the gauntlet was thrown down, and the phrase "come at me like you want to kill me" was unironically used. Kirishima commented about how manly it all is. Midoriya looked moderately concerned murder is going to be committed down the street from where he lives. Iida looked like he wants to somehow be the responsible guy and intervene, but didn't know how. And on the inside, Bakugou knew that Todoroki was probably regretting for the umpteenth time ever associating with him, but feelings are feelings and pride is pride. He is emotionally stunted, but has enough pride in his left pinky as most people do in their entire bodies.
It makes dragging Todoroki over to his house a somewhat more awkward affair than usual, because Bakugou has expressed a desire to casually kill or be killed at some point (literally dragging, as in he clamps onto the tail of Todoroki's blazer and refuses to let go of it in case he tries to stage an escape from his fate, which would be kinda cute except it's Bakugou). He gets the feeling that Todoroki only lets it happen because there's a part of him that wants to believe Bakugou is just kidding, or will be persuaded out of this really fucking bad idea.
He will not be moved by logic. As far as he is concerned, his home is merely a pit stop to drop off his school bag before they go attempt to beat the shit out of each other, only just letting go of his deathgrip to do just that before turning around and giving Todoroki an expectant look. ]
So.
[ A dramatic pause ensues, to build the suspense, or something. ]
You're not gonna half-ass it, right?
[ Please Bakugou, he probably hasn't even agreed to this stupid idea yet. In which case, Bakugou would be doing worse than property damage. He would be performing straight-up homicide with intent to kill. ]