You seem well aware of the first way to control or automate an app, which is using keyboard shortcuts that the app builds and documents in its menus. Most apps are good with this, but even Apple apps sometimes force you to move and click the mouse to do things (eg, Find My.)
The second way to control or automate an app is using AppleScript. Most apps have a mechanism that allows users to control the app using AppleScript commands. But as with keyboard shortcuts, many apps are missing some of the methods needed to control the app this way. In order to see how much support your app has for AppleScript, you can check that yourself by using a macOS app called Script Editor and opening the application file itself in Script Editor. Since I don't own the app in question, I can't do this. And even if I had the app, I'm not very adept at writing solutions using AppleScript.
The third way to control or automate an app is using KM actions which read the screen, analyze the screen's images or text, and simulate keys and mouse movements in order to automate the things you want to do. This is usually a complex methodology, but when it works, it works. I do this all the time to solve problems. For example, on Saturday I noticed that the Apple Find My App doesn't support either the first or second approach for controlling it, so I wrote a KM macro that reads the screen, clicks on it, and sends me the text of my friend's addresses to my iPhone via SMS. I had to account for the position of the Find My window, the locations of my friends' names in the sidebar, and the location of the inspection window on the right side when it pops up. It sounds complicated, but once you know how to do it, it's not that hard.
I trust that you've correctly eliminated the first approach as a possibility. We don't know if we can eliminate the second approach yet.
One thing you haven't indicated is how you perform the requested action right now. Do you use the mouse on the screen to achieve this? Are you manipulating a control on the screen? Can you find a video online (or can you create one) that shows how you are currently doing this?