Definitely go for an adult social care referral and urgent OT assessment. It can be arranged really quickly, but try to get it for when you're around as if your dad isn't able to respond they will probably ask your mum questions.
My dad is still competent and said he was happy to use incontinence pants rather than a hoist to get him in the commode. Within hours of OT leaving he wanted to be put on the commode.
What I have found is that any professionals who are involved will put the patient's wishes over and above absolutely anything else, including the practicalities and logistics of family members involved in caring. This is probably the right thing in theory, but when you're the one picking up the pieces in between carer visits ita bloody frustrating.
More importantly it sounds as though it will be your mum that's having the input as your dad isn't able to, and as though you can't really trust her to act in your dad's best interests.
Its a horrible situation to be in. Does your mum have any friends who might be able to talk her round?
I am in a very similar position, dad unable to weight bear, and very frail, possible cancer but can't undergo invasive testing so no conclusive diagnosis. He is largely compos mentis, apart from the odd spell of confusion when tired but has difficulty communicating.
Mum wears the trousers but is highly anxious, in denial and full of conspiracy theories (we had to have community nurse administer a morphine injection last night which caused an almighty tantrum). She doesn't agree with him having 'too much' pain relief and keeps insisting that all would be fixed if he could only see physio a day magically be able to walk again.
I am the frazzled carer you speak of in your OP. For your own mental health as much as your dad's safety and wellbeing, get as much support as you can however you have to go about it. It's really not a fun place to be.
Also don't be afraid to keep 'pestering' the organisations that can help. It's very easy to get sidelined if you're not persistent.