The solution somewhat depends on what the other players are using the "downtime" for.
If they are all pushing for short rests between combats to use their hit dice for healing, then a Cleric of the Life Domain is particularly good. In terms of mass healing for cheap, you might consider a single level of druid, or two levels of Ranger, with remaining levels put into Cleric. Why the Life Domain cleric? At first level, that cleric gets "Disciple of Life" as an ability.
Whenever you use a spell of 1st level or higher to restore hit points
to a creature, the creature regains additional hit points equal to 2 +
the spell’s level.
Why the Druid level? The spell Goodberry will produce 10 berries, each of which (because of the Disciple of Life cleric bonus of 2+1, see above) will heal 4 hit points rather than the usual 1. That's 40 hit points to distribute to your party, for the cost of a single first level spell. At 8th level, you'd be able to see to most of your parties healing needs with just this combo, unless your DM decrees that eating that many Goodberries is bad for their digestion.
For mass healing, there are a few spells, but no lower level ones are as powerful as the combo I just mentioned.
- Prayer of Healing is also an efficient way to heal a group (2nd
level, and especially effective when cast by a cleric of life).
The bard's Song of Rest is effective as a healing boost, but requires a short rest, which is what it sounds like you're trying to avoid.
Beginning at 2nd level ... any friendly creatures who can hear your performance regain hit points at the end of the short rest by spending one or more Hit Dice, each of those creatures regains an extra 1d6 hit points ... extra hit points increase ... to 1d8 at 9th level, to 1d10 at 13th level, and to 1d12 at 17th level.
For higher level spells, there's Mass Cure Wounds, which a 12th level
character could cast 3 times a day. Your Druid can already do this,
but it doesn't hurt to stack another on top of it.
Your Druid could also cast Conjure Woodland Beings: if your DM lets
you chose which particular fey are summoned, then the Druid could
summon two to four Dryads, allowing you to get 60 to 120 hit points
These various healing abilities should reduce the need of long rests between encounters.
One last thing to consider: your party might not just be taking so many rests to heal. Several abilities regain their uses after a short rest, and your party might be counting on that.
For example, your Moon Druid will want the rests to regain uses of Wild Shape; your Eldritch Knight might want to regain uses of Action Surge. If you do decide to go Life Cleric, you'd benefit from rests as well as you regain your uses of Channel Divinity. None of these can be regained instantly through a support class's abilities.
Although a healer is likely to stop the party from needing as many long rests, short (one hour) rests are still likely to be quite popular, even if everyone is at full hit points.