Poly can be very soft, and can handle tension.
That said, YYE Poly isn’t particularly soft.
Nor does it whip that well. At least not by comparison to many other strings.
The “thin side” of things limits your choices. You would probably need to go with something based on trilobal thread, which will take away from the “softness” factor. It’s a major balancing game, and I don’t think you’re likely to hit all those bases. Might be time to pick your two most important ones and go from there.
Quick 13’s T13x is on the thinner side and whips amazingly. It’s not “rough”, but it’s also not particularly soft… it’s just got that plasticky trilobal feel to it, so it depends on your definitions of “soft/rough”. Tension… well, it’s not anything special. Not terrible or anything, but there are better strings for managing tension.
So there’s an example for you. Thin yet whips well. Had to sacrifice some softness and tension management to get’r done.