Straight from the DMG Errata:
Attunement (p. 136)
The first paragraph ends with a new sentence:
“If the prerequisite is to be a spellcaster, a creature qualifies
if it can cast at least one spell using its traits or features, not
using a magic item or the like.”
emphasis added
Therefore, for the purpose of attunement:
If a creature is able to cast a spell by virtue of their own ability (not from an item), whether that be race, class, or feature, you are a spellcaster.
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
of goodberries.
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.
Best Answer
There are a few radiant options open to your party.
Moonbeam, a 2nd level Druid spell, is available to the party Druid. It lasts for a minute, concentration, so it's a good way to keep the Radiant damage going without spending a spell slot every turn.
Spirit Guardians, a 3rd level Cleric spell, is another spell that deals radiant damage repeatedly if your Cleric can maintain concentration.
Look for a Sun Blade for yourself or the Eldritch Knight. It's a rare magic sword with the finesse property that deals Radiant damage.
A Mace of Disruption, another rare magic weapon, is also an option. Its only drawback is that it only deals radiant damage to undead and fiends. If you ever need radiant damage against a creature of another type, you'll be out of luck.
A Zealot Barbarian (from Xanathar's Guide to Everything) deals additional radiant or necrotic damage on the first attack they hit with on each of their turns starting at level 3. This may not work for your party, as it would require some multiclassing.
The Paladin spells (also available to War Domain Clerics) Divine Favor (1st level) and Crusader's Mantle (3rd level) add Radiant damage to your weapon attacks and the weapon attacks of all allies within 30 feet of you, respectively. They each last one minute and require concentration. These are similar to a "lesser Holy Weapon", but may not be easily accessible to your party.
It's also worth noting that the Daylight spell that you mentioned ironically doesn't create sunlight (see this question), nor does it deal radiant damage. It wouldn't actually be able to help you with a vampire. If the spell did create sunlight, it would say so, as in the 6th level spell Sunbeam.