tbh i think bowser is the "nicer" partner and would bring flowers to eggman
I can imagine Bowser would try to be a real romantic like he tries to be with Peach and it'd be cute when he tries to do something nice for Eggman like giving him flowers and stuff. but Eggman would ruin it constantly by not being into it at all most of the time because he's the exact opposite and doesn't have a romantic bone in his body, his approach is just having a friend he can do evil and mischievous shit with and also be a fuck buddy
so he'd often be like "haha lol what is this gay shit" to Bowser's advances if he goes beyond that with special things he tries to do for him and Bowser would pout like "come on stop I'm trying to be sentimental here". though he'd be in luck sometimes if it's not too intense for Eggman's liking and makes him feel special and adored because at the same time he's always looking for that attention and boost to his ego. and Bowser will take what he can get, a win is a win lol
Bowser is also the sort of better father overall and would more commonly have more genuine cuter moments with Jr but neither of them are that great because they're chaotic and Eggman is especially the worst father ever. and they'd totally exploit the way people would just find them being two dads coming together and their kids together cute anyway and use that to get away with many schemes
I find their dynamic and all the possibilities very funny to imagine XD