Hi MH_UK - I realize you are in a tough situation and wanted to say best of luck and I do hope your problem gets resolved. As someone else mentioned, you have the option of maybe going to a lawyer and having them provide some insight into this as I am sure they would have encountered many scenarios and this could have been one in the past - At any rate - you will have some professional advice and if the lawyer makes you feel comfortable, maybe you could hire him/her to do your mom's application process.
The other comment I wanted to callout is that at least in Ontario, it IS possible to be separated and live in the same house and if this is recognized here, there MAY be a way to prove that this is the situation in your home country as well - You could perhaps get a court document that is attesting to this and that may save your mom the embarrassment of having to physically leave your dad as no one would know other than her kids/close family members but that court document should be considered as a legal proof of separation, even by IRCC.
This is just my 2 cents here and hopefully more experienced contributors can provide insight here if this may work.
One final thought that just struck me is that one argument to be made for your dad to consider doing his part in helping your mom's application move along is the carrot that she'll leave and be with you and he'll be free from her company. Wishing you all the best!
