Ironically, I think the PvP healers make a bit more sense from a design/identity perspective, despite the limited moveset/similar core.
For example, PvP AST is the fast and mobile buffing healer, with instant cast times on their single target GCDs. They essentially have something similar to Red Mage's dual cast trait, letting them instant cast a GCD after a hard casted Benefic. Their speed technically makes them the strongest throughput healer. PvP AST doesn't have any shields or any regen, save for drawing lady of crowns through it's job mechanic. Their card buffs, speed/mobility and throughput healing is what they are given in place of shields or on-demand regens. They do still have Disable/Bole for occasional mitigation though. Not having shields actually makes drawing Bole a bit more desirable as well. (Pray for Bole on limit break)?
Given that AST has abilities that involve time manipulation and speed alteration, I think it makes sense thematically as well.
The lower recast time on Lightspeed and the instant cast trait of PvP AST makes it so the developers can comfortably omit certain healing tools on AST's kit to create a more varied feel. AST doesn't require the same regen/shield mechanics that a WHM/SCH has when it can heal so rapidly. I feel as if PvP AST finds a third way when it comes to keeping the party survival potential that each class brings relatively balanced.
PvP WHM is more of an omni-healer rather than a pure healer. They still don't have utility, but when it comes to survival, they can do it all (shields and regen). They're given several very strong oGCDs that they need to keep people alive. The lily mechanic also makes way more sense in PvP. You need those oGCDs ready as much as possible due to having slower GCD heals. 3 Lilies gives a full 50% cooldown reduction. It's a really strong effect compared to the 20% reduction you get in PvE. Some situations leave no choice, but due to the healing intensive nature of PvP, you definitely don't want to pop WHM oGCDs before 3 lilies if you don't have to.
Of course, fighting other players is very different. You're healing way more often in PvP, so things like personal DPS and utility are a bit less important. It's not clear how well they would translate, but the best bet may be a long-form version of the PvP healers for WHM/AST.