Garthanos
Arcadian Knight
True but there's only so much you can do. I think a fighting style that aids every round that is balanced against existing fighting styles is probably not going to have a big enough impact to be felt.
Fighting style are not very high impact in 5e. So there is cross purposes I agree. TWF seems to be potentially getting your strength once more per turn? So allowing a mental attribute bonus on the first and follow up attack (could be analogous), but now that I think of it since the TWF is forgoing some defense. I think only the first following attack against the same enemy gets the benefit.
It would combine well with my scan for opportunities (skilled superiority) where the fighter can forgo an extra attack to gain a superiority die which could make the rallying into a bit less once in a while. You arent just inspiring fortitude but bolstering performance.The other thing it would do is interact with the Purple Dragon's Knights ability to give out temp hp to allies with it's second wind. It would do a considerable amount to improve that particular subclass as it would enable it to set up a nova strike round at an early level.
Heroic exertions that let you accept damage for a bonus to skill use? and allow that to eat through temporary hit points was also a related idea