It's possible to run a command in another tmux pane; I forget the specific syntax but I've tested this in the past and it works. The only thing the editor needs is a task runner (this is a planned feature for Amp); the task that's run is prefixed with the tmux command that triggers it in another pane.

The only thing that's bespoke to tmux is knowing which pane to target; you'd need to collect that from the user on the first run, or allow the value to be configured.