Wise up OP! I think from what you are posting that you are understandably overwhelmed with a situation not of your choosing. Fact is though you are in charge of making sure your DM is being cared for appropriately and not being taken advantage of.
I have 2 decades of experience both being a carer in other people's homes and employing carers for family members. Everything about your posts rings alarm bells. The main carer is not being professional, not being appropriate and not behaving in the patient's/(client's) best interest.
Your mother is not well, she has a degenerative disease and caring for her should be about getting professional expert assistance when needed. Any professional carer worth their salt should know this, so;
equipment query - OT first port of call every time, OT will perhaps add physio, try out centre, equipment reps etc
food choices/ compulsion - Dietician referral and maybe GP for blood tests combined with redirection to good healthy diet options and behaviour management/distraction techniques to avoid binge eating etc
spending choices/ compulsion - hobbies/outings/socialising to redirect - basically more behaviour management/distraction techniques to avoid binge spending etc
If the 6 cashmere jumpers in a week or all out tantrum is no exaggeration why isn't the carer at worst just rewrapping (with your agreement obviously) already bought brand new items for the dopamine hit without the waste? This is not about controlling spending it is about symptom management. Why isn't the carer helping tidy up the wardrobe with your DM to show what she has already - put together planned outfits etc for the same activity with zero waste?
Being a carer is not just about doing what is needed. Being a carer for a physical disability is about levelling access and doing what is asked whilst encouraging independence, resilience, strength etc. Being a carer for a cognitive disability (particularly a degenerative one) also involves more guidance and patterning - nudge psychology if you will. This is how the job of ensuring healthy medication/exercise/hygiene etc routine is done happily, by building rapport with the patient.
Your DM's main carer is nudging in the wrong direction. Add to that involving you in agency management, manipulating your guilt and the encouraging of distancing tensions within the family and you have a problem.
There should never just be one paid carer, it is encouraging of an unhealthy dynamic (as you are discovering) and creates issues if someone is ill or not being professional.
I would swap the carer team and if you haven't done so you really ought to check the receipts for stuff bought compulsively against what is still in the house. It may be possible to return unused items and, because your theft focus seems to be on money - which you rightly state would be obviously missed, however things in a cluttered shopaholic house? not so much. Are you checking the supermarket receipts or just looking at the total?
Of course the carer is lovely, (she couldn't guilt you about money by being anything else!) But lovely doesn't absolve lack of professionalism and in worst case could be a smokescreen for taking the piss. You said;
I genuinely don't think the carer is scamming and we have full oversight of my mother's accounts and she tends to run expenses by us. She's not a bad person, far from it. I think she likes being the facilitator for the fleeting joy my mother gets from spending money and getting deliveries.
but this is not what a professional care for someone with dementia looks like, this is not encouraging fleeting joy in active, productive or creative pursuits. It isn't even taking her on an outing to the shop to get her out of the house is it?
I like the main carer who makes my life easier by taking a lot of initiative. But god she loves spending other people's money. So much smoked salmon! She's always on at me for something - the most expensive Dyson, a new bigger TV etc. We are in control of my mother's finances but carer can use the card and also asks for stuff. I don't think she's dodgy, but she's deeply irritating. We have POA. It feels like we're being put into a tricky position when we're asked for stuff. I would have liked to have been asked before the chair company came round - we just got sent photos of DM looking gleeful in the 'deluxe' model and told she'd already chosen the fabric.
You are being manipulated. What happened to the old dyson or the old tv btw? Think, even overlooking the total bypass of OT consult, why would any carer purposefully set up a sales rep meeting for a compulsive spender without inviting you along? Because you are being set up as the challenger instead of the understandably concerned relative.
Good luck OP, please keep in mind that your DM is altered and hopefully you will both have a better chance of getting along better with more appropriate care in place that encourages your relationship instead of distancing it. Beware that this carer doesn't become a bad penny type "friend" if you replace her.
Sorry for the essay, just trying to illustrate from one what has been there too, stick with it! If it was me I would employ carers directly (not easy but you can pay a good rate, do your own interviewing and checks and supplement with another agency in the meantime) and I would change the locks and be around a lot during change over. In my experience of working in the care industry 70% are there doing a great job with heart and soul, 30% are there because they don't have many job options - your description is not heart and soul variety.
If you are not already in touch with local carer support groups get on their lists, there are always free online courses being run on stress management, disease progression and support, employing directly and dealing with this sort of situation.