Grace
Since you already have him in a hospital cage with heat, can you place a white paper towel or white paper underneath him on the bottom of his quarantine area, take photographs of his droppings, and post them here in the forum? It may help the members narrow down what exactly is bothering your poor finch.
I often notice that if I give a bird an antibiotic like amoxicillin, they get worse before they get better as the medicine kills off all of their gut bacteria, even the beneficial strains. Since you are already giving him antibiotics, keep an eye out for a possible fungal or protozoa infection flaring up, since the good gut bacteria helps keep some of the bad things lurking in check.
If you can, you may want to consider ordering a probiotic to give him after his treatment with amoxicillin is finished, assuming what your bird has is a bacteria infection, and not something else.
The one thing I'm noticing that is probably not directly related but might be contributing to your finch's condition is that from your list of foods that you've mentioned, I'm not seeing anything that is rich with vitamin A -- zucchini has a little, and apples have even less.
Anecdotal story time!
Once, I was purchasing a fawn cock from a family who unintentionally bred their pair of zebra finches and were trying to rehome the offspring. While I was there, I noticed there was a hen who looked like she was just going to give up the ghost right there. When I asked the person who was negotiating the sale about her, he just gave me her as a "freebie" since he suspected she was going to die anyway, and didn't want to figure out how to nurse her back to health.
The hen had had diarrhea, loss of appetite, wheezing, bad feather and beak condition, lethargy, clicking sounds when she breathed after flying too much, etc, but it turns out rather than infection, she was suffering from a vitamin a deficiency because she was fed a seed only diet in her former home, and she wasn't eating any of the dark green vegetables I offered to her since she was very suspicious of new foods. Eventually, I had to supplement through her water since I didn't have any powdered vitamins on hand, but after two weeks she had a noticeable improvement and has long since made a full recovery to a gorgeous full bodied hen.
I'm not saying that Vitamin A will cure an infection, but a vitamin A deficiency can compromise a bird's immune system, so it is something to consider while your get your boy back on his feet.