Ah! I see. That hidden field idea is tried and true.
We also found a solution to my original question with an answer so obvious that everyone probably thought we knew it already. (.submit (.getElementById js/document my-form-id)) does the trick of performing the same action as clicking a submit button. So, with some careful handling of the form-id to pass it around appropriately, we can perform the submit at the time we want to, after the other thing.