Running an AppleScript using BTT shortcut integration?

I want to create a shortcut that will toggle bluetooth on, set it to my AirPods and randomly play my recently added songs on Apple Music.
I want to set the correct output using an AppleScript. Is there a way to create a BTT without any specific trigger (keyboard, mouse or otherwise) that I can summon using "Execute named trigger" in the shortcuts app ?

Apple's Shortcuts app can run AppleScripts from within them.. It also supports shell/bash commands.. So I'm not seeing your need for BTT if you're wanting to use Shortcuts to initiate the actions. All you're wanting to do can be done directly in the Shortcuts app.

However, you can have BTT active a "Shortcut" via multiple of options..