A melee healer is one that has only melee attacks and is considered a melee for boss mechanics. An example is the Mistweaver Monk in WoW, who would never be targeted by boss abilities that are meant to keep the ranged moving (puddles, spreading debuffs etc). If there are any such mechanics that specifically target healers (like in A4) then the melee healer simply doesn't do any dps during those mechanics. There are both positives (not being targeted by certain mechanics) and negatives (not being able to dps in every situation) to being a melee healer so I'd consider it balanced overall.
I think it could be a healer that can actually take the spot of the SCH, unlike Nocturnal AST. A healer that has significant dps and can cause automated healing during dps, but whose "pure" healing capabilities are not as good as a WHM or Diurnal AST has. To make sure SCH stays put they could make the new healer (I'll call the new healer DISC) incompatible with SCH in some way. For example make DISC dps cause random absorb shields which overwrite the stronger SCH shields. When balanced in such a way the best comps would be:
WHM / DISC
AST / DISC
WHM / SCH
AST / SCH
Everyone would have an equal chance to get a raid spot, and have 2 good co-healer options and 1 not so good option (WHM / AST and SCH / DISC). Nocturnal would remain the way it currently is: a stance that is used only when paired with another AST (or in some cases WHM) in DF.