Are there any lifts that allow a patient to be lowered and raised into a bathtub?

Yes, if you can get your mom onto a chair that is in the tub, you can buy a portable bath lift chair (rechargeable) that works great! I've even used the chair to raise my mom off the floor after she has fallen and is unable to stand back up. I have her scoot her butt onto the chair and then I use the chair to lift her up to as high as the lift goes, and I can help her stand from that seated position.

Do go to the top of the timeline on AgingCare and use the search bar to search for "Best Lifts". You will find a host of questions about lifts for all sort of needs. You might go to medical equipment strore in your area and check on rentals as well. We often get guestions about lifts here, and I hope today someone is using one ideal for this purpose and will communicate the brand to you. In hospitals where I worked we used Hoyers. I have zero idea the cost of such a lift. If you find the best one do write to us here as sharing information on good finds is how we learn from one another.
I wish you the best of luck.