What is the immediate, next step, after 911, for a client who can't be transferred with assistance by one caregiver while in her own home?

The AL needs to know your client is a 2 person assist and cannot help in transfers. ALs don't usually take this kind of assist. Seems to me your client's family needs to know this. She needs Longterm care. You need to tell them until they get a place for ur client two caregivers are needed. The people that are sent when u call 911 will eventually tell u they can't continue to come out. Your client needs more care now.

The next step is to notify your supervisor that this is an "UNSAFE" assignment for 1 person.
I would notify them in writing as well as a verbal notice.
The agency should then notify family that this is now a 2 person job.
If the agency refuses to acknowledge that you have stated that this is unsafe then you give your notice. (Or state that you can no longer work with this particular client on your own)
In some cases even with 2 people it is unsafe to do a 2 person transfer and appropriate equipment has to be used.
(Even though in a home equipment can be used by 1 person I think agencies would require 2 people. I know once I had to switch from a Sit to Stand for my Husband to a Hoyer the Hospice required 2 CNA's)
And no you can not constantly call 911 for transfer or Lift Assist.

The "immediate next step" should be you giving your 2 weeks notice, and letting the family deal with their loved one accordingly.
Like said, 911 won't continue to keep coming out just to help you transfer this person, as that is not their responsibility. They are there to help if someone has fallen and needs assistance getting up.
You are stuck between a rock and a hard place and it's time to get out now before you hurt yourself.

After calling 911 medics will help with a transfer, but I’d doubt how wise or needed any transfer is for a person who isn’t able to assist and needs ongoing two person assists in a home setting with only one caregiver. That’s usually called bed bound, with no getting up. Medics will not repeatedly come do this without either charging a fee or triggering a report of an unsafe living condition, depending on where this is. If they transport to hospital there will have to be a medical reason the person is kept there before release to either rehab or assisted living. But assisted living does not provide two person assists for transfer as a rule. As a hired, private caregiver being put in the position you are, I’d call 911 and have the person transported to the hospital for evaluation, give the medics the name and phone number of relatives, and back off from there. You’re being placed in an impossible position, completely unfair to you and the person needing care

The family of your patient is who you should consult since they're handing the arrangements.