Sure but it gives the "optics" you aren't clean. Which we all know she doesn't care about, but he's a teenage boy.
Asking to go home to bath is just as embarrassing though. Not sure how homie thought that was better.
He and Sumi are at that akward, overly self-conscious age, when suddenly everything is turned askew. While previously using the same bathwater, or even bathing together, was innocent, now it’s… not.
There’s intimacy and familiarity and trust implied that he feels is inappropriate for their age and current relationship. He does not know and cannot be sure how Sumi will view it, in particular with their age appropriate and more polite addressing each other. For instance on the negative end of the sprectrum, she could be disgusted with the idea of her sharing a boy’s "dirtied" bathwater, or in that same vein would not want to have a boy get dirty thoughts with her bathwater—which has, after all, been all over her.
Add to that respective crushes, and the japanese incliniation to avoid any offense and default to non-offensive alternatives and offense avoidant ambiguities, and it’s pretty clear.
He offered to take a bath at home to avoid the awkwardness of her sharing his bathwater, or he sharing hers (depending on who goes first).