The sensei are immortal, incapable of negative emotions, have no warp presence and have mysterious "good warp" powers in other words they are horrible mary-sue saturday morning cartoon characters. They are incapable of grimdark or character depth and in my humble opinion are a bit too "light and hope" at odds with the death and despair of modern fluff.
True. But, as GW maintains, "there is no canon". Nothing requires you to recreate the Sensei exactly as they were (we never actually saw any of them in the Jaq Draco trilogy, anyway, only heard them described, and pretty much everything else that the Harlequin Man feeds Draco is at least suspect and at worst knowingly a load of outright fetid dingos kidneys).
A more recent analogue (from the Horus Heresy series) is the Perpetuals. Who I know a lot of people still aren't a fan of but who fit the universe a lot better.
Essentially, it doesn't matter.
as for personality, psychic powers and states i'm at a loss.
A big feature of the Perpetuals (which I approve of) is world-weariness. Dying and coming back is a horrible, sanity-wrecking process. As is watching everyone you know age and die and forever having to move on and move on to hide who and what you are. This guy will have lived for millenia, died probably several dozen times and keeps coming back whether he likes it or not. He'll probably have tried to kill himself at least once and found that doesn't work either.
Psychic powers...could be anything. If you want him to be immune to the touch of the warp, then just ignore psychic phenomena, but you might decide not to. After all, if 'immunity to the warp' was an inheritable trait rather than a facet of personality, mind and soul, you'd have thought the Emperor would have had the sense to include it in the Primarch genome.
One thing you might do, even if you don't want him to be a psyker, is to make him a sorceror. No-one can live that long, especially around reaver crews and the scum of the universe, with both Chaos and the Inquisition intermittently chasing him, without learning a few 'tricks'.