IME, complaining probably will achieve results for getting her showered more often, however (and I don't say this to make you feel guilty but because it's reality) it will mean someone else not getting showered, or other people not getting supervision etc they need. The management will simply detail the carers to shower more often and 'just manage'. That's my experience of this kind of situation.
Even without covid, it's unlikely they'd be happy with you helping out by giving her a shower, not necessarily because the care staff have got anything to hide, but because of the risks associated with using the stand aid and equipment and needing to be trained and covered by insurance etc when delivering care, plus the exposure to other residents.
WRT things that can help, I second sending in plenty of shower gel, a sponge or two (I find they're softer and they hold more water when giving someone a bed bath than a flannel) plenty of underwear, and dry shampoo/the caps - care homes usually have hairdressers that go in but some don't at the moment and that's something else that then falls to care staff, I used to love spending an afternoon setting a couple of the ladies hair and painting nails, putting handcreams on - unfortunately needing to give basic care such as toileting and feeding has to be the priority and so the nicer things fall by the wayside with staff shortages.
The home should have beakers with lids and handles for residents that find it hard to use a cup or mug, but if you Google 'beaker cup for elderly ' there's a lot of different ones available to buy - put her name on anything you buy in permanent marker so it doesn't wash off, it's impossible to keep track of every single persons belongings by sight without names and unfortunately some residents can be magpies.
Unfortunately if there's not enough staff, then there's not enough staff and those working have to balance the risks of all the residents in their care, they can't defy the laws of physics and be in two places at once. The CQC regs state that there have to be enough staff to meet the needs of the amount of residents they have, in number and qualifications. Unfortunately the CQC are slow and clunky, and seem very reluctant to use their powers when standards fall short, and even they can't pull more staff out of thin air. Staff are reluctant to speak up because there are no real safeguards against retribution against them when they do, and if they do, very little if nothing changes anyway because the regulating bodies are about as much use as a chocolate fire guard in most cases.
An already short staffed sector just got made worse (whoever you choose to blame for that) and will continue to get worse as the pressure the ones who stayed increases and they are blamed for the failures and they leave too because it wasn't really worth the money paid before, it's certainly not now, and you can work in an environment where it doesn't affect people's lives and well being for the same or more money, being in that situation takes a real mental toll on the carers too.
I don't know what the answer is, obviously speaking up is a start - but is anyone actually listening to the whole story? I can see a headline about something like this along the lines of "90 year old gets a shower once in 3 weeks!" And lots of exclamations around how bad that is, and the CQC saying "We're investigating" and the providers saying "Lessons have been learned and heads have rolled!" But will there actually be more staff available to provide the care? Or will it continue the way it has with the staff on the floor getting a bollocking?
I know which one I've experienced and I left because despite the investigation and me and others telling the truth - nothing changed.