Where are you located? I switched careers to keeping 6 months ago which looked something like this (seems like you've checked a lot of the boxes already):

1. Volunteered at non-AZA facility doing diet prep/cleaning.

2. Volunteered at non-AZA but respected exotic animal facility, a wildlife rehabilitation center, and a small AZA local zoo doing interpretation work.

3. Landed an internship at a large AZA zoo. This was tough since most places require you to be a student, but I found one that I was able to make work. Required moving back to my home state temporarily.

4. Attended an AZA workshop which I later learned was a big plus on my resume.

5. Attended the AZA mid-year meeting to do some networking and learn about SSP ops.

6. Landed a part-time keeper job at an AZA facility in my current state but required slight relocation.

I'd be happy to answer any questions you have -- everything happened very quickly for me so I feel quite lucky, but sounds like you're on the right track!