Paladin as a tank / healer in combat is possible, but inefficient.
Building a Pathfinder character is all about tradeoffs. By mixing roles of frontliner and healing support, you can't really specialize at both effectively; you output less damage than a dedicated attacker, and restore less HP than a dedicated healer. Support casters often struggle in the front lines, because they're either busy attacking (i.e., not healing) or are taking too many hits.
Paladin is an ok choice, although their healing and support role is secondary to their melee power. Paladins are suited as a tank in combat, and a support/healer out of combat. However, they can still manage the combat medic role if planned and played strategically, and it sounds like you're set on pure paladin.
Usually, healing is best performed out of combat. Early on, a wand of Cure Light Wounds will be helpful. Eventually you'll want wands of higher-level paladin spells like Lesser Restoration.
A paladin's healing/support capacity improves at medium levels (5-10, usually). Combat healing in Pathfinder is suboptimal, because HP is typically lost faster than it can be regained. Therefore it's better to prevent incoming damage, by engaging and defeating enemies, than to spend your turns as a band-aid. There are exceptions (e.g. Heal) but paladins don't reach that level of spellcasting.
Combat tactics:
Buff and heal sparingly, as needed. Your spells per day are very limited. Outside of melee, you should rely on wands and scrolls.
Don't wait for the enemies to rush your party. Move into tank position as soon as possible. If you expect a difficult fight, maybe cast a buff spell for yourself and/or allies on round 1.
Prioritize tanking over healing. When you're in melee and a nearby ally is getting low on health, the enemy is your immediate threat. Deal with the enemy first, then assist your allies.
Encourage your allies to carry healing items (e.g. potions). If your allies can heal themselves via their own action economy, then you can spend your turns more effectively.
Useful feats:
Power Attack. Useful for many tanks, especially with 2-handed weapons. Select your tank and frontliner feats early, since you can't cast at low levels anyway.
Reward of Life. Make your Lay on Hands more effective, because Paladins have more daily uses of Lay on Hands than spell slots for healing spells.
Combat Casting. Although 5-foot stepping away from your opponent is often the easiest way to cast spells, sometimes it's not enough (e.g., your enemy has natural reach) and you'll need to cast defensively. Concentration checks are tough for a Paladin because their caster level lags behind the primary casters, and so a +4 bonus will help.
Selective Channeling. Tanky characters tend to be near enemies. This feat lets you channel positive energy to heal your allies without helping your opponents.
Archetypes:
I recommend none. Many of the paladin archetypes replace one or more of your Aura class features, which is a mediocre tradeoff. Because your Aura abilities are passive effects, exchanging one for a standard action ability hurts your action economy. Also, since you're going to level 20, your party will need reliable saving throws versus the onslaught of save-or-die effects; they benefit more from the Aura's bonuses.
You don't have a party balance problem unless you create one.
In certain editions of D&D (mostly 4th), there were definitely more stringent requirements for class/party balance. That is not the case in 5th Edition. Any party can be successful, if they play to their strengths. As a DM you always need to pay attention to the types of enemies you throw at the characters.
What's challenging for a melee heavy party may be simple for your caster-heavy party, while the inverse can be equally true. You should also pay some mind to what the players like. Some enjoy mind puzzles, others enjoy intricate tactical combat, some enjoy pure socializing. As long as the DM doesn't put up deliberate roadblocks, forcing the party to us a single solution, your group should be okay.
Character Competencies
You do have some misconceptions about how the bard and cleric can function. They are certainly both capable healers, but they don't have to focus on that. In 5E, bulk healing is often an out-of-combat task: heals-per-round can never keep up with damage-per-round. It can only bring people up who went down and maybe buy a little extra time. Good crowd control is actually more important than good combat healing, and crowd control is something both the bard and cleric can do.
Depending on domain, Clerics can be competent at healing (Life is super good at it), battle (War or Tempest), or even combat spellcasting (Light). All clerics have access to some real gems, like Guiding Bolt (solid damage, and grants advantage to a later attack) and Spirit Guardians (one of the best lower-level area control spells).
Bards can heal, sure, but honestly... there are a lot of classes that can get access to healing. What bards really do is support and control. Don't underestimate Bardic Inspiration; a few points here or there can be clinch. Even low level spells like Dissonant Whispers have good crowd control abilities, causing people to run and provoke attacks of opportunities. Hypnotic Pattern, an excellent lock-down spell, is on the bard list too. Furthermore, any bard can potentially learn any spell in the game - Magic Secrets is no joke, and Lore Bards get it relatively early.
The only one-trick pony in your group is the warlock, and even that's a slight exaggeration. They're certainly less flexible than other spell casters, but warlock invocations can be very powerful. If the player picks the right spells (if it doesn't scale with level, it's probably not a good Warlock pick) they can be a powerhouse.
A Note on Worldbuilding or "No plan survives first contact."
Never forget that it's a gaming group. The DM's responsibility is to facilitate fun for everybody. It sounds like part of your fun includes the world building aspect, and that's great. Just be careful to give the players options; let them explore your world, don't force them down a particular path just because you have a neat bit of lore to share over there. Encourage, but never force.
Best Answer
The Healing Light feature gives you a bonus action which basically replaces the Healing Word spell — which you don't get.
You're going to be leaning heavily on that, because your selection of healing spells is limited — the warlock doesn't have any on its normal list (except some necromancy stuff, which only heals you), so all you have are those from the Celestial Expanded Spells feature: Cure Wounds, Revivify, and Lesser and Greater Restoration.
At 6th level, it is probably going to be fine. You're missing the "big guns" like Mass Cure Wounds and Heal, but those aren't available to anyone at this level. And fortunately, you've got bards for the Song of Rest, so your party has short-rest healing benefits available.
I don't think you need to do anything particularly special to "build" this — just avoid focusing on other things that will consume your bonus action for Healing Light. You could look at the Healer feat, but in most cases going for straight Charisma boost is going to do more — it increases the number of dice you can spend at once on Healing Light, and significantly improves all of your non-healing stuff, while the feat just gives you a weak alternative to the Cure Wounds spell you already have. The Acolyte background would give you story-based access to temple services, which might be handy depending on the game and setting — but it sounds like you already have a background in mind.
For a Charisma-based divine caster, you may look instead at the Divine Soul sorcerer — you could generally flavor this to fit many similar story ideas, and you'd have access to the full cleric spell list (and twinned Healing Word is awesome).