I think I managed to find the appearance of 5H4D0W-TP

Borderlands: The Pre-Sequel BL: TPS Gameplay Discussion (Spoilers!)
#1

In the Arena. I managed to nail it down. I think.

He appears every 4 times for me.

If I lose/win 3 arena fights, the fourth time 100% has him appear in the third wave. Does anyone else have a similar experience?

Winning or losing does not matter. Just play the arena 3 times.

#2

I get him to spawn pretty often on the first try on difficulty 4 after save/quitting.