She then asked where I'd got it from when I told her
it had come through the post because of the POA...
Ouch.
Seriously, that was extraordinarily insensitive of you. Basically she now knows that you ARE prying, and that is a terrible feeling.
Don't get me wrong, I have POA for my mum, and her spending is so out of control that she could be out of money completely by the autumn. My brother and I both have POA, but we have to be really empathetic and sensitive about how we talk to her about what we find out about her finances. She's a difficult person, but over the last six months she's agreed to various strategies that we've suggested, that will mean that she can still have (some of) her treats in a years time - never mind having money to feed herself.
But we have never put a statement in front of her, and any mention of her balance etc is almost made in passing, and we talk more about how to plan for the future and how much she has coming in and going out. It's extraordinarliy difficult to balance the conversations, as she still has capacity, like your mum. So she could easily go back on some of the things she's allowed us to do (like look after her bank card and just bring in an amount of cash every week - this can only happen by agreement as we have no right to keep her bank card)
Putting myself in my DM or your DM's place, seeing a paper statement and discovering that it was sent to you, really brings home the loss of control that a POA has. And at this point you can only act with her agreement, so you've put the whole thing in peril.
You really need to have some empathy for her. I have an LPA with my DDs which of course they don't use at this point. But theoretically they could, with my agreement, and if they did what you did, I can only imagine the shock of the realisation that I had no privacy or control.