Fire custom submit handler in form embedded in node edit form
I came across https://www.drupal.org/forum/support/module-development-and-code-questions/2015-09-03/put-a-custom-form-inside-node-form which, while old, claims that what I wanted to do is not possible. To get this to function the way I wanted I:

Attached a JS event handler to the embedded form submit buttons that performed .preventDefault()
This handler then .ajax'd a custom route that I created.
In the Controller for that route, I had to perform all the primary operations for submitting the form and return markup back to the .ajax call.