Do warlords need to be equal in raw HP gain to other healing classes?
I don't think there's a solid number, but comparable, sure. Enough to be the sole support character in a party without disadvantaging them in that area.
By the same token, the player would need the flexibility to divert some/all of whatever resource is consumed to extend hps to other purposes, in case the party composition shifts, and just to be able to adapt to the challenges of the day.
Or could they be weaker on raw HP gain, and superior with indirect HP gain, like bonus AC/DR/THP?
I think it could vary wildly based on build or even player choices in play. In fact, I think it already does, in general, so it won't make much difference.
When you think about it, 5e is very inconsistent when it comes to hp resources. If it were just hp & HD, you probably couldn't make it through the requisite 6-8 encounter day. But there are multiple classes that can extend hp resources, and others that can't. A designer can't know what every party composition will be, some might have little ability to extend their hps, others a lot. Even a the party contains one or more characters with such abilities, many of them could be used for something else - a Cleric or Druid could prepare no healing spells at all, for instance.
You could presumably calculate what a class adds in terms of maximum hp extension, imagine if a cleric cast all his slots as Cure Wounds for instance.
A new class would only be problematic if it greatly exceeded the range of damage mitigation already available to other classes.
All that sounds like the kind of thing that would be of concern in actual design, and maybe fine-tuned in playtesting.