First off screw your brother, it's not his call and i believe unless he has some sort of legal court granted power that you on your own can give permission for your Mum to have a biopsy.
secondly you have done nothing wrong and everything right, and should be proud of yourself, not feeling awful. this sort of care is exhausting to anyone and you need proper respite which csocial services should be able to offer you.
Please make a HUGE fuss about her not eating... my ex wife's grandmother literally starved herself to death and none of the many medicals that my wife called in did ANYTHING to help her or her family.
But most importantly please remember that you have done nothing wrong, and you deserve to feel good about yourself.