I think you are doing the very best thing you can do in the circumstances.

While I would generally side with those who believe that a cat with sores on her chin, no collar and who wanders the neighborhood indicates a caregiver who isn't giving proper care, I have to remember one of my cats that would have given the same impression.

I shared Sam with a neighbor. I provided vet care, food and shelter that they'd refused him, but they still insisted that he belonged to their children. He'd always been an outside cat and loved children, so I allowed him out in the daytime when I was home. He developed chin acne, a really bad case that had him hospitalized for a few days, and it took weeks of antiobiotics and twice-daily chin scrubbing to clear up. Anyone seeing Sam then, and I found out that everyone in the neighborhood with children knew him, would think he was well-fed but not properly cared for. He lost collar after collar, but he was micro-chipped. He would eat up a storm at home but beg food from every neighbor and act pathetically underfed.

I hope all goes well for "Rosie".