I would try a futaba or mks tail servo. I fly my N7s with 106mm tails and 14.5 degrees of collective and I cannot blow the tail out period. tail servos make all the difference in the world.
with the neo, tail gain is different now. the number you set is not really the exact gain. it has some sort of programming that senses for drift and also oscillation and it will change the overall gain. so when tuning the tail on the neo now, we use the expert menu, because that is a concrete number that doesn't change on its own.
116mm tails should be perfectly fine too, maybe 5 lower on the p gain than the 106mm blades and they should be perfect.