Ugh..been there. It sucks.
As for your seatbelt solution, you can get one of those bath tub chairs.
And there are these visor type things that you can put on LO's head so that when you pour water, it doesn't go in the face/eyes.
DD is like this too, but we just keep on trucking. I take a lot of baths with her, DH takes a lot of showers with her after dinner (since she likes to comb food through her hair every night), but I'd say with all the shared baths with me and showers, she takes an equal amount of baths on her own (we don't want to get her used to just one way and have her not cooperate). So, when she needs her hair washed, we lay her back, she fights whoever, and we just make sure to help her out by watching for water in the eyes and drying them off when we can. We do try to give her toys for distraction when we lay her back to wash her hair.
We also have one of those detachable shower heads that hang down to the bathrub and we can use that on her too.
Just looked on amazon and they have this 'portable shower hose' that you can slip over the bath faucet to help spray LOs hair. If you got the visor thing, and the chair, you might be set. lol. Lots of contraptions out there to get the job done!