Unsure of what to do with dry gangrene in finger

My mom is a double amputee (LAKA + RBKA) and has had gangrene in her toe that required amputation at one point well before the leg had to go.

I spotted a cut on her middle finger that turned black and brought her to our emergency room because I was hoping to avoid delaying care. I thought they amputate the tip and she would be healing by now. That was 12 days ago. They instead kept her overnight and sent her home without any instruction except to see a hand plastic surgeon in 4 days.

The hand surgeon passed us off to his PA who said to schedule out 14 days based on the notes from the hospital.

I am frustrated because now the necrotic tissue has all fallen off, leaving bare bone sticking out. She is not 100% with it and just thinks its bazaar, which it totally is! The healthy tissue below the demarcation line is now all red, swollen, painful. The pain started several days ago and I think thats why the originally hospital sent her home, she didnt have any pain. But now its different. I remessaged her PCP to get her take on this as well. She generally sides with me and understands my moms complex issues. I feel like the hospital didnt really care or want to do anything about it but kept her overnight and sent her off.

Not sure of what to do here. I feel helpless and now think this will lead to losing a finger instead of a tip, and she will likely need IV Antibiotics which give her hallucinations.

I was trying so hard to prevent all of this, and I worry it could take her hand. I dont know who to call or if I should bring her to a different ER and expect better care. The one I brought her to isnt the best, never acknowledges her DPOA on file so if i leave to sleep i have zero clue of what they did to her until i can grill a nurse.