What support is available for a caregiver when they are getting burnt out?

I think you are a saint, many would have already left her. She is fairly young, maybe get her meds re-evaluated, perhaps something is off or not working for her anymore. Yes, get her family involved as they can take some responsibility for caring for her and give you a break.
Who has Power of Attorney? if the answer is no one, then you need to discuss this with them and decide.
Does she have insurance? if so, she could qualify for home aide that can come in and help her out during the day while you work, because it sounds like she is disabled enough( she may have to have a doctors referral)
It sounds like you have a lot on your plate and asking her family for help is not a bad thing. If her family is willing to help, take the help and at the same time talk to them about long term care for her and what that would look like, or put a more permanent plan in place to have them help you moving forward.
Care giving without help equals killing yourself slowly. Take care of yourself!

Are you your partner's PoA? Is anyone?

Below in a response you wrote:

..." I am partly to blame for things getting to this point by letting her cancel therapy and doctor appointments."

Unless she doesn't have legal cognitive capacity, you can't prevent her from cancelling her appointments, so don't blame yourself for trying to get an uncooperative person to do something they don't want to do.

She has a complexity of mental and physical health issues. Is she on opioids for her neck/back pain? If so, is it possible she is abusing those meds? Or underdosing herself so that her pain is not controlled?

If she's in a shelter, you may want to have a discussion with staff there (if they are receptive) to give them the lowdown on problems. This may be an opportunity to get her into the ER or a psych wing of a hospital to get things under control before she goes back home where, if nothing changes then nothing changes. Adding her Father to the mix of caregiving/enabling may seem helpful but might not be in the long run.

Also, it is very heart-warming to know in your mind she is your wife. I just want you (and her) to know that the states do not share this thought and that excedingly few states recognize "common law" marriages. This may or may not matter if she needs to qualify for SSDI or Medicaid, if she's not already on these. Mostly she needs to know that even if she stays with you for many more decades, upon your passing she will not necessarily be treated with the same legal status as a wife. This may mean she cannot apply for your SS benefit, and may not automatically inherit any of your assets unless she was already joint on the account or named on the title, or as beneficiary, etc.

If she cares about you at all she will assign you as her DPoA so that you can legally help her in whatever way she needs. Leave the daughter our of it... it's no longer appropriate to expect or ask her to help, even if she seems willing. She needs to go live her life.

One more thing: at 43 your partner could be entering peri-menopause, which can be messy for women:

"Mood changes and irritability are usually due to a combination of sleeplessness and hormonal swings. Some women find the menopausal transition to be psychologically stressful; some develop clinical depression. Women may be more at risk if they have experienced severe PMS mood swings or have a history of clinical depression. Perimenopausal depression usually goes away within a few years after menopause. In general, depression is less common during the postmenopausal years than in the premenopausal ones."

Source: https://www.mountsinai.org/health-library/report/menopause#:~:text=Perimenopause%20can%20begin%20in%20some,be%20sudden%20surges%20in%20estrogen.

Since she's not legally your wife, your say-so about her care may be nil. Did she grant you her POA? Or does her daughter have POA? Or has no one thought about that so far?

Your caring for her is admirable, but as for your legal responsibility - you probably have none (unless you are POA or guardian or somehow have other custodial rights).

Pay a visit to an eldercare lawyer. Usually the first consultation is free. Sort out the legal ramifications, and the lawyer may have advice about how to find her more care as well.

As for the daughter, don't expect much from her. She deserves her own life, at last, and shouldn't be roped further into this tangle. Neither you nor her daughter are to blame for the fact that this woman canceled health care appointments and didn't take care of herself as she should have.

I hope you will all find the help you need.

Sound to me wife may have Dementia. So hopefully she will be tested during her stay. I would say split your assets but your not married so legally you aren't obligated to care for her. So hope you have not c-mingled your money. Time for the daughter to find her way. I would place your "wife". If she has no money, Medicaid may pay for it but she is so young. Do not use your own money. And I would transfer her from the Crisis Center right to a nice LTC. Then get the SW to help you get her applied for Medicaid.

care.com , Next Door.com Has people advertising to help elders or you can Place a ad on Next door. com , see if there are any Nursing schools near by also .

I don't understand.
You are caregiver.
How does the stepdaughter leaving the home (which is what she SHOULD do for the sake of her own life) have to do with your wife now going to a 10 day crisis care care? Is it because you are required to leave town for work?

Whatever the situation you now have the opportunity to meet with the Social Workers at Crisis care and ask them to assist you in explaining to your wife that you cannot go on caring for her in-home and working.
She needs placement now with this level of care.

Asking a daughter to give up her life to do this care is not right.
You need also to see an attorney about working on division of assets so that your own future is protected. Your wife will be self pay for some time I am assuming.

You badly need the intervention of social services, and you need to come to an honest understanding with your wife. Only your fear of pain and grief is preventing this, but there is no way around pain and grief in this situation.

And please, encourage daughter to stay out of this care for her mother. Let her get a life.