You would have to write a completely different menu from scratch, in either C or Vala (Python support will be removed come Budgie 11, so I wouldn't invest your time in that). Arc Menu, as with all other GNOME extensions, is written in JavaScript / GJS.
You can use Budgie's existing menu as a reference point and since it's an applet like anything else (including third-party applets), the structure (like registering with libpeas) is the same.
Generally speaking, I'd suggest looking at Vala's Tutorial to start learning Vala, Valadoc for documentation on APIs in Vala, etc.