Warning - long reply 
First thing before anything else i would compose a letter to the GP that you feel would be best, make an appt even if it is weeks away to go and discuss it as then you have an agenda and you both are on same page at start of appt without you having to explain anything, bring a copy of the letter for yourself to refer too
in the GPs letter state all your concerns, including the risks such as perhaps not being able to clean properly after using commode (even with help due to embarrassment) and the potential for skin to breakdown etc, what you fear would be the long term consquence of not taking any action to prevent this and discuss solutions to prevent things going down this path,
By any chance are you registered at GP surgery as carer? even if you arent, they cant tell you anything about your mum as she is an adult however they can take on board what you are telling them and your parents may be off the generation who would do as the dr advised, even if was the same as you but it holds more weight cos the dr said it, if you get my drift
i second what other pp said about involving district nurses, im 90% bedbound due to 5 different conditions, i would be lost without my hospital bed and other equipment. I was resistant at first but for sake of not letting skin break and causing excess pain i had to put pride to one side, best decision ever. I also have 4 care calls per day
I can understand a son not wanting to do personal care tasks for his mother however he has taken on the role of carer and this doesnt allow for picking and choosing what parts, he will or wont do.
"I do their banking, sorting out doctors, collecting pills, food shopping so I do contribute" im not suggesting that you withhold care or lie to get a care package but if you were unable to do these things would your brother step up to the plate and take this on, or is it only selected parts of caring again?
re Banking can be done online, you could help them with this via a second named card for the account, and then you have monthly statements and you make sure all bills are paid as and when needed on time.
re Sorting out doctors and collecting pills, pharmacies can be good if a person is on a long term regime of the same meds to set up a re-ordering service so that they will re-order on the patients behalf as needed and put the meds in time slots in a dossette box. Some pharmacies also have a "drive thru" pick up window, others have delivery services so check out what is available in your area. Make things easier on yourself.
Food shopping as you know, could sit down with a mobile wifi connection and do a grocery shop etc with your mum / dad, then brother could take delivery and put away. It would allow your mum to make the choices of which 1 of out x number of choices she would like, same as in a supermarket, bringing that back to her as she may have always done her own shopping and might miss that now she is housebound effectively
Are all pathways in the house clear? for your mum with her walker as last thing she needs is a fall,
"Am I a bad person for wanting bathroom facilities for my DM? I'm feeling very hurt by his comments that he is the one doing everything when it is not true. Ok I moved out and have my own house and kids, but I work full time whereas he has never worked in his life".
No you are NOT, definitely not, from the way i am reading it and please correct me if i have picked it up wrong, he is doing meals and hoovering and as a "carer", he isnt actually doing any direct "personal care" tasks at all - due to embarrassment or otherwise
Brother may be aware that if he is not a carer anymore, he will be required to sign on and look for work.
"My dad helps but he is not in good health but he empties the commode which my DB wont touch." - :( - what i see from is this,
he is living rent free, some housework, getting "carers" allowance which i conceed isnt much for what a lot of carers do and majority out of love, he is basically like a live in housekeeper/carer who doesnt do personal care which is not what your mum or dad needs
"Your dad will get to a stage (soon I suspect) where he is no longer able to do these things for her. i suspect this will come soon", also you need to try and speak to both of them seperately so they can give you their own thoughts on this, your mum is in danger of a fall due to weakness, sores from lying in an unsuitable chair as a bed and infection from same sores or from pressure points anywhere else
" I am going round 4 times a day/night" - if you couldnt do that, they would prob need to resort to outside help at some point, ask for a care assessment and explain you cant help all the time as its taking a lot out of you (and im sure it is)
You mention that your mum would not move without her dog and cat - have you heard of The Cinnamon Trust - The only specialist national charity for people in their last years and their much loved, much needed companion animals. A network of 15,000 volunteers “hold hands” with owners to provide vital loving care for their pets. We keep them together - for example, we’ll walk a dog every day for a housebound owner, we’ll foster pets when owners need hospital care, we’ll fetch the cat food, or even clean out the bird cage, etc.
re washing when i havent been able to do a proper wash, i can suggest the following
Re hair washing - no rinse shampoo cap - feels like a real treat
body cleanser - senset cleansing foam as it leaves me feeling really fresh
good luck and if you want to ask me anything as ive been the carer, now the caree and also the daughter trying to get a stubborn parent to accept care 
and 