My little boy has started doing the same thing, he absolutely loves bathtime and he likes to he fairly submerged so laying him flat probably wouldn't work for us. I'm hoping for suggestions too
Depends how much you bath them. I often just get in with mine and hold him. I can play a lot better and also either prop him up between my legs or lay him on my legs (to practice for swimming). He won’t be (as) scared if you have a positive reaction to him slipping. Smile, laugh and give him cuddles like it’s a big game and then go back to it. He’ll gain his confidence off you.
My daughter never liked the seat so I've been laying her in a bath from day 1 😅 she splashes a lot! (she had a small plastic bath)
When my daughter outgrew this, we either laid her flat in the bath with a small bit of water or used the next bath seat that's a ring- although they need to be able to hold their head. My son has started doing the same as yours with that baby seat