Playing FGO on TV Questions

So i just bought a Typc to HDMI Adapter and connected my Samsung S9 to my TV to Play some FGO.

Seems to work fine, Picture looks clean and FPS is good (stutters way more visible here though)

Anyway, i can only control using the phone as a Touchpad and wondered.
What if i bought a Bluetooth Mouse and hooked up to Phone to play with it?

Has anyone tired it this way?