I had to move mine to the big tub around that age, because she would stand and jump around in the little tub.
Bought a bath mat. She stood in the big tub...so I'd just wash her quickly, and take her right out with no toys. She eventually sat down.
Then, this summer, I took her down a huge waterslide on family day at the pool, and my feet slipped when we landed and we both went under. I don't know if that traumatized her or what...I had a tight grip on her the whole time, but we were probably under for a good 4-5 seconds (felt like an hour, it was the scariest event I've had with her) but ever since she won't sit in the tub. She doesn't mind at all if I dump a bucket of water right over her head, she still plays, etc, but she just does not want to sit down.
I've started bribing her with bath toys...I don't give her the toys unless she sits. I did win the battle just yesterday, and she sat for the first time in about two months. Yay!
They have special bath seats that suction to the bottom of the tub. I had one that I loved for my first...it looks like a cross between a potty chair and a restaurant high chair, I guess, with a ring around their tummy and a bar between their legs. Worked great for her...she wasn't a great sitter, either. We never even tried a bath seat with this one, because she just didn't need it.
You can lay a folded bath towel down in the tub...yes, you'll have a sopping wet towel to deal with afterwards, but it offers more protection than a bath mat.
You also might want to consider having the sliding door removed from your tub and replacing it with a shower curtain. I think the door is going to continue to give you trouble...that'd be a real pain in the rear when it came to bathing a baby or a toddler.
Good luck!