Hm, the idea seems nice but some I just wanna share out my thoughts.
No primary for the soldier is a huge deal as not having the ability to get away or do medium/close range damage hinders soldier. Also look at what you're giving up for the melee slot like the garden, escape plan, and the disciplinary action. Knowing that, I think the equalizer needs to be a good weapon but only for a niche use.
The main loadout would be the equalizer and battalions/conch as the buff banner and the rest are useless unless the Equalizer has a speed on hit which I don't think that needs to happen. Since the Equalizer has reduced damage on higher hp I don't think the banners need changes. It takes 600 damage to charge a banner (480 for the conch) so a soldier needs to hit the boss 7 times with full hp (98 damage on full hp with crits) but just needs to hit the boss 2 times when you have the most minimum hp for the most damage so something like high risk for high reward and even then, it takes time to cast the banner leaving you vulnerable.
Heal on hit is fine as long as it doesn't make soldier invincible if he blindly fights the boss just like the pyroknight. The damage reduction should only let soldier tank 2 hits even with/without the battalions. But you might say that having damage reduction and the battalions effect is op and to that I say is, the boss can just jump away from the soldier and let the banner go to waste. The battalions in this case would be a last resort option or a scare tactic to the boss. And the rest of the changes are fine. Additional optional stuff: Maybe a longer melee range idk.