Any way to throttle calls to a specific API in Chrome DevTools while leaving others unthrottled?

If you can change the URLs, there's a site http://www.deelay.me/ that is a delay proxy for HTTP resources. For example, if you want URL http://example.com/example to be delayed by 5 seconds, change it to http://www.deelay.me/5000/http://example.com/example.

No, Chrome DevTools network throttling does not allow you to do selective throttling.

If you want to simulate specific services being slow and you have them running on your local machine, I would recommend installing a throttling proxy (e.g. Charles proxy).

If those services are deployed and you still want to simulate sluggish response, I would try amending /etc/hosts to point to a local charles proxy as well.

Someone has now built a chrome extension for this - URL Throttler.