What happens if AL can't handle her issue?

Maybe you can get Visiting Angels to come to her and help her shower and take the clothes and put them in the laundry area. Introduce Visiting Angels as your friend "mom this is my friend - Alice she will be coming and the you two will be doing A and B then C. (Not letting her choose should help.)

From the responses you have given, it’s definitely time to move to memory care…

The next step is memory care or a nusing home where she will have a lot less freedom and independence.
The AL has other residents living there so the place is not going to tolerate hoarding and filth in one of their apartments because it will attract bugs and vermin. People get evicted from apartments when they let the place go because it becomes a health risk to the other tenants in the building.
If the AL has meals in a common dining room no they cannot allow a person who smells and wears dirty clothes because the other residents have rights too.
Memory care or a nursing home will force her to be cleaned up. They will clean her room whether she refuses or not.

Memory care without a doubt. They have all sorts of techniques to coerce a reluctant person to shower and stay clean.

I don't know what they did with my mom, but she, too, wouldn't shower when she was still at home. My dad became distraught because she was overweight and smelled terrible and he still had to share a bed with her. It finally fell to me to shower her, which was no picnic either, but she fought me a little less. Once she moved to MC, she was clean and shiny all the time, and I'm sure a lot of it had to do with their facility and the existence of shower chairs and so forth.

Memory care is absolutely not only for wanderers. Its focus is less medical and more on mental stimulation for their residents, they have a far better caregiver-to-resident ratio, and the care is infinitely better -- at least that was our experience. My mom started in a skilled nursing facility where she was largely ignored because she wasn't sick enough to warrant much attention. I moved her to MC, and her world changed completely. The surroundings were more like an actual home, the people were nicer, and she wasn't only approached when someone wanted to give her a pill or give her a shower.

Some good info on showering, here:

https://youtu.be/VXO8peuarEs

If your moms assisted living facility "can't handle" her hygiene issue then yes, memory care would be the next step.
It sounds like you may be in a bit of denial about your moms condition, as you say that just because she doesn't wander or is not combative that she doesn't belong in a memory care facility, while you also say that her cognitive decline is "progressing rapidly" and has no short term memory.
It definitely sounds like your mom is ready for memory care. Hopefully the assisted living facility she's in now has one that she can progress to. If not have their social worker help you find one for her.
There are so many reasons that a person with dementia refuses to bathe or take a shower. Often it does have to do with fear. Fear of falling, or fear of the water hitting them,
It may be too that she's thinking she just had a shower and is forgetting that she hasn't since her short term memory is all but gone.
There does come a time when living in an assisted living facility just isn't feasible anymore for a person with dementia, and that time has now come for your mom.
Best wishes in getting her place in a memory care facility.

Does this facility have memory care? What do they suggest?

I am sorry that you are dealing with this situation.

There does seem to be an issue with bathing in so many elderly people.

Is she afraid of falling? Is she cold? Do you know what is causing her resistance to bathing and wearing clean clothes?