Should I insist on a sponge bath/hair wash once a week?

Well, based on the info you've given it sounds like your mom is deeply depressed. No, it would not be acceptable to me. Get her to a geriatric doc and have her tested for UTI, etc. My aunt had started to stay in bed (not anything like you are describing) and after putting her on antidepressants, multiple supplements, changing her diet, strengths of her meds etc., we discovered her thyroid meds needed adjustments. She was full of poop even though she said she went every day. (Took two ER visits with x-rays and a trip to a specialist to get to the right doctor and treatment.) This was her situation, your moms is probably different but I'd have her get a thorough workup. I had to change doctors to get my aunts taken care of. I think her old primary was depressed himself. Sometimes, depression can lift on it's own but your mom needs help IMHO. If you can't get her up, call for an ambulance. Also check into PT to get her going again. It really helps.

Physically there IS something wrong. Sorry to say it but you are in denial about her health. Get her to an MD or start picking out the flowers.

If your mom can't or won't get into a shower or bath tub, then she should at least get bed baths - at least 1 bed bath or shower per week, whether she's depressed or not. Try getting an aide (an HHA) to come to the home to do this - an aide might have better luck getting her to go along with the cleaning. If you can't afford an aide, try giving her bed baths yourself. There are a lot of CNA training videos on YouTube that demonstrate giving the elderly/bedbound a bed bath (among other tasks). I found them helpful when learning to care for my own mother. She's bedbound, with Parkinson's and dementia. Showers were awful when she was still able to get out of bed, but things became easier once she became bedbound. The aide and I both use baby wash with washcloths and basins of warm water, but you can get a rinseless shampoo/body wash for adults. - we use that to wash her hair. They're not great, but they're good enough. Make sure the room is warm enough for her before you start on the bed bath. Cover her with a large bath towel or bath robe and only expose each body part as you wash it, to keep her body warm. I'd also recommend not using the washcloth to clean the nooks and crannies in the rear. Use disposable wipes, warmed up in clean water, no soap, then toss in the garbage. That stuff is a pain to clean out of a washcloth....and get her to a geriatric doctor. She might be depressed, but this might also be an early symptom of something else.

One more thing: I also clean under her breasts each morning. It's a high perspiration area. If not cleaned and monitored, a bad rash can develop. If she develops a rash there, do not use any diaper rash creams to get rid of it. They will only make it worse. If she gets a rash call her doctor for advice on how to treat it.

All good advice and I hope not too late for this elderly lady, but just to point out that the original post is six months old.

Does she have a hospital bed? A portable commode. You can't say there is nothing physically wrong at 81 years young. The body loses muscle mass. My mother is 80. I have a hospital bed to adjust her, elevation where it's easy for me to care for her. Her bed is next to a portable commode. I set an alarm to go off on the hour for her to check herself, wearing pull ups. She can stand, pivot and sit on commode. It is a foot away from bedroom door. On back, I have a shoe rack which has her supplies. She uses baby wipes versus toilet paper. Also helps to keep her clean. Everything goes in the garbage can next to the commode. Small bags from Dollar Tree in the can. I have to make it as easy on me as the caregiver and keep her as independent for as long as I can. I now give her meds ground up in applesauce. I make sure all food is easy to swallow. 81 is a great age, but muscle mass is a given.

Although this is a six month old question i am sure there are others out there who are facing the same problems.
I would suggest a bed bath at least twice a week or more often if the patient is large or sweats a lot. Paying particular attention to skin folds and the peri area which should be thoroughly washed every day. When some one is incontinent that area need to be washed with plain water or soap and water or a wet wipe every time they void or have a bowel movement.
it is very difficult to say when someone is lazy, abusing their caregiver or really dealing sith an expected decline at age 81. If possible they should be encourages to get out of bed at least for short periods and if possible walk a little even if it is only round their bedroom. This helps with breathing and circulation which in turn helps prevent bedsores.

As far as washing the hair is concerned, it really depends on how much hair the patient still has left. If someone is bald or has very thin hair it might be possible to simply wash the head but for fuller locks some kind of shampoo is needed. I personally hate those shower type shampoo caps. They are easy to use and warm up nicely BUT my hair felt horrible afterwards.

There are cape like contraptions that allow you to use real water right in the bed. if your loved one can still leave the house a visit to a beauty parlor once a week makes a good excuse to get them out of the house.

Sitting out of bed for meals if possible helps digestion and also helps prevent aspiration especially if there are swallowing difficulties. Each situation and patient are unique so where possible try to follow the patient's wishes but don't forget your common sense. 

Mom may hate a shower because she gets chilled but welcome a good wash down in her own bedroom, Just make sure to dry each part as you wash it. The bath tub can be frightening but maybe your loved one will tolerate being washed down stilton a shower chair. A walk in bath tub is another great option if the funds are available for installation.