HB - I've not really got involved with this before as it's impossible for anyone outside a situation to understand the dynamics and the pressures. But enough is enough.
Contact Social Services and your mum's doctor and fully explain the situation to them - tell them, in no uncertain terms, that you CANNOT do anything physical to help your mum, but that your mum is being awkward and won't allow 'strangers'.
Calmly pass on what is said to your sister, remind her (as subtly or unsubtly as necessary) that you physically cannot do it, that your back is precious and if you don't take care of it the present situation will become permanent.
Then step back. You say your hubby is supportive, but less than subtle when dealing with people who just don't 'get' it - so let him deal with them for a while. If you mum kicks off, let him tell her how it is, if your sister gets unreasonble, pass him the phone.
My dad had a laminectomy and 4 vertebrae fused when he was 32 - the surgery worked and he was mobile until he was taken from us by cancer ... but he drummed into me a respect for back pain and back care that few around me seem to share - too often, niggles and even severe pain are brushed off as 'nothing' and people who take time off work to allow back injuries to heal are called malingerers. You CANNOT play fast and loose with your back - listen to what it is telling you.