I agree, I think you are dealing with early Salpingitis as well. Take note of your photo - that "bean" - it's sitting on
top of the poop and looks relatively "clean" (not mixed with fecal matter), so for me and my thought process, it came from the oviduct instead of the intestines. Further confirmation is your description of "cheesy".
I would treat with the Amoxicillin. Dose is 57mg per pound of weight given orally twice a day for 10-14 days. She weighs 2lb 2oz so that comes to 121.125mg - soooo 125mg twice a day is what I would give (there's wiggle room in amoxicillin dosing).
Do you have tablets or capsules? The powder in the capsules does not mix well with water, so it's better to mix it into something like coconut oil.
For example if you have 250mg capsules mix it with something "known" like 1 tsp coconut oil. Mix well, divide that in 1/2 - there's your 125mg dose! You can roll that coconut mixture into a little pill shape if you wish, put in the freezer to harden, then just pop it into her beak like a pill. Make sense?
I would also treat the crop like you are doing. It may be slowed due to infection - crop issues and reproductive disorders just seem to go hand-in-hand. So working on both conditions, hopefully she will start to recover for you.
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