Well, that console message shows a couple of big clues.
1. It’s not the PUT request that saves the API Token that is failing, after all. It looks like it’s actually the subsequent POST request that queries your kits after saving your API Token that is failing.
So I’m guessing that you see the green check mark next to your API Token, like it succeeded in saving the token, right? Even though it then subsequently fails?
You could troubleshoot to make sure that POST requests are also working correctly on the plugin’s REST API routes. Another place this happens is when the plugin does the “preference check” when you switch configuration options on the “Use CDN” tab. Before saving, just when you click around to select different configuration options, you’ll see it spin and then report whether saving those configuration changes would result in some conflict. That preference check request, that happens before you click to save the changes, uses POST. So if you wanted to make sure your web server was handling POST requests correctly, you could inspect that the preference check request is succeeding.
2. The WARNING might indicate that there’s some other code–plugin or theme–running on your WordPress site that is undermining the Font Awesome plugin’s functioning. I’ve seen that happen when other plugins emit output that they shouldn’t during the process of a REST API request/response. It’s why I added that whole diagnostic there that this WARNING is talking about.
If you want to look into what that might be, you could inspect that request in the Network tab of the JavaScript/web/dev console. Look at the contents of the response. In the cases like this that I’ve seen before, there’s been something in that output that provides a clue as to what other plugin is undermining the REST request/response.