I can't see any way round it, other than your DM saying clearly and repeatedly that she can no longer provide care. SS will continue to take her at her word if she keeps saying she can.
FIL found it very hard (for different reasons) to say that he couldn't care properly for MIL. He was just rubbish at it - didn't seem to notice that she was not washing or toileting properly, kept leaving her to dress herself (DH turned up once and she had a cardigan wrapped round her waist as a skirt - FIL said he'd told her to go and get changed and seemed bemused that she hadn't done so 🙄). He'd give her her medicine and expect her to take it, and wouldn't check, then he'd find a stash of pills hidden in a hankie, and he didn't think it really mattered. SIL was run ragged trying to keep up with the domestic side of things - to which he was oblivious.
But he was determined to keep her at home as that's what she said she wanted when she was asked.
In the end DH and SIL coached him to say that he couldn't have her home from hospital as he couldn't keep her safe, he couldn't provide the care that she needed at home. And they jointly stuck to that line when SS wanted to send her home (with osteoporosis, immobile, doubly incontinent, Parkinsons and dementia) with daily carers. She went into a nursing home. However, if FIL had cracked at any point and said he'd have her home, SS would have jumped at it. Even now he doesn't get it - still talks about bringing her home with no thought of the reality of it. In the home she has company, assistance, is fed and kept clean, takes her medication properly. She's far better cared for there.
Anyway, maybe you could take this tack with your mum? That your father is simply not safe at home with just her? That in doing what he says, she's actually doing him a disservice - even if he doesn't see it that way.