Advice for a 1 year old who will not sit on the bath unless she's in a bath seat (then she screams) or she's happy but standing and falling down around the tub

I'm at my wits end. Bathtime has become a nightmare. My daughter has a bathseat to keep her seated, but she fights it and throws herself around. We bought the bath seat because otherwise, she happily stands up and falls over constantly, even with you right there trying to keep her sitting down. She's slippery.

The other day she whacked her forehead and got a goose egg, and tonight she bit her tongue pretty badly. But, she's happy (but getting injured).

If we put her in the bath seat, she throws herself around and throws a tantrum and will whack her head off the side of the tub (we have to pad it with towels). We tried sponge baths in her room but that was even worse.

I'm just at a loss. Bathtime has become a nightmare! We constantly tell her "Sit down" and end bathtime when she doesn't sit down. She's 12 months and I feel like my choices are either have a happy (but injured) child or have a safe but miserable child who hates baths.

Update: I think I'm going to try bathing with her. We have a large non slip mat but when you're already unsteady on your feet like she is, it's easy to slip and/or fall. As soon as she stands I take it as bath time is over anyways, so I quickly rinse her off while saying "All done! Standing up means all done!"

Thank you everyone for suggestions :)