I’m going to mix metaphors here, but your current situation is a “house of cards” with you as the cornerstone keeping it all shakily up.
Considering human nature, of course your dad is happy to have you keeping the cards standing. Why should he consider placement or extra help as long as you are willing to do what needs to be done? He’s tired at 86 and you’re the daughter, probably makes perfect sense to him!
You and you alone can decide what you can humanly do and what you owe yourself and your nuclear family. Placement may not be an option if your dad is firmly opposed, next step is arranging for/increasing private duty help at home- but your dad will probably resist that too, “strangers coming in”, “too expensive”, “why do we need that?”
Last thought- people in your mom’s condition are very susceptible to pneumonia, if this unfortunately happens and she is admitted to the hospital, the discharge planner (RN or social worker) will contact you pretty immediately after admission. You can tell that person all that’s happening at home and they will help you look at options- going home may be considered “unsafe due to family is unable to provide the patient’s required level of care”, which is true- you need 24 hours of rested caregivers to provide 24 hours of care.
I admire the many posters here who are taking care of their loved ones at home, but not all situations allow this. My vascular dementia husband declined to being super-volatile, profanely shouting insults and threats, refusing meds and meals, switching day for night. Family was afraid to come, I was afraid too- barricaded the door to the gun room with lots of heavy things, as he had hidden the key. I moved to the guest room and locked the door at night. Then he left the house at night and rang a neighbor’s doorbell, the neighbor brought him back. The next day, his PCP squeezed us in, sent us to the ER, they admitted him, 2 week stay- lots of tests, bloodwork, neuro and psych consults. I talked honestly to the discharge planner, he was discharged to assistive living and is doing SO much better with the structure of the environment and the help of 24 hour staff care. I am doing so much better too.
There is a lot of talk now about Work/Life balance. Think about things and don’t feel guilty about whatever you decide, but be aware- Dad will be happy to have you continue what you’re doing, so you will get pushback for any change, especially those that cost money. Could your husband be any help in talking to Dad about the effects on you and your family of what you’re doing and the need to give you some relief?

Usually it is financial. It is my understanding that Assisted living is charging $5,000 and up and Nursing Home Care is $10,000 and up. This is monthly and there are extra's like having someone do their laundry or if you take more that the number of pills that are allowed under your monthly fee. Has you dad though of hiring someone to come in an assist with your mother's care. Unfortunately, the care in a Home will not be the same as you can give her at home. You did not mention how old you are. What will your father do if you move? This is one of the most difficult times in a family. It sounds like you all could use a little counseling. Have you discussed this problem with your doctor? I wish you Good luck
Gina5009