Not sure if this made a difference, but I changed the PUT URL to drop the .json and now everything seems to be working.
To figure that out, I started up Fiddler and made a change directly to my profile in Discourse. I was able to see how the form data was sent and it was exactly how I had it, but the URL was slightly different.
I’m sorry to waste the time of everyone that read this, but it is a good lesson learned for me. ALWAYS look closely at the little things (and monitor the network to see how it is supposed to work).