Prioritize number of attacks and Feat combinations
Ok, a quick addendum before we go any further. There is no such thing as one perfect build for every situation. The environment you'll be fighting in, the type and number of opponents, and even the nature of the story you're engaged in can all change which characters are "best" suited. For example, a character who hits once but hits extremely hard (like a rogue) may have higher DPR (damage per round) than another, but against a horde of weak enemies (e.g. Goblins) they may find themselves less useful than their DPR would suggest they would be, since a large amount of their damage each round is wasted hurting an already dead enemy.
All that being said, I've done a bit of research and found that at this particular level, there is a build that will work very well in a lot of scenarios, and meets the criteria you've set (survivability and DPR without any other players' help) in a measurable way. So enough hedging: let's look at the details.
The Build
Stout Halfling (counterintuitive, but hear me out), Eldritch Knight Fighter 13, Rogue (or Wizard) 2
Str: 8
Dex: 20
Con: 16
Int: 14
Wis: 12
Cha: 8
AC: 18, 20 with Haste, up to 25 with cover
Feats: Sharpshooter, Crossbow Expert
Favored weapon: Hand crossbow
Favored Spells (in order of importance): Haste, Mage Armor, Shield, Find Familiar (or another 1st level Wizard spell), 7 more evocation or abjuration wizard spells of your choice (you'll rarely be casting them).
Rationale
In descending order of importance, the three factors you need for maximum ranged weapon damage dealing are:
- Multiple attacks
- Accuracy & Damage Bonus (about equal: both come from Dex)
- Base Damage (weapon's dice)
Multiple attacks are almost always the most important feature. A typical enemy in DnD will have an AC which can be hit on a natural 8 or better. So usually, an increase an increase in accuracy (say, the 2 point increase from the fighter's "archery" style) will grant you about a 16.666% overall boost in expected damage (hitting on 14/20 attacks rather than 12/20), while a boost in number of attacks will grant you at least a 20% boost in damage (going from 5 attacks to 6, which usually can only happen at 20th level anyway). Although enemies with particularly high AC may alter this math, on average multiple attacks will be of more benefit. That being said, we're going to get both as high as possible.
With all that in mind, you're going to want the following build:
- 13 levels in Eldritch Knight: 11 for 3 standard attacks. At level 13, trade in the spell you gained at 8th level for "Haste", and get another single weapon attack (4 per round total so far).
- At level 4, take crossbow expert feat. Now, according to sage advice (http://dnd.wizards.com/articles/features/sageadvice_feats), you will be able to use your bonus action for another hand crossbow attack. Although the hand crossbow has the lowest damage dice of the martial ranged weapons, this feat makes it the most damaging in most circumstances. (5 attacks per round by level 15).
- At level 6, take sharpshooter. Now, you can shoot your hand crossbow just as well up to 120 feet (and since most combat in a dungeon crawl happens within 100 feet, you're golden). Also, as long as your target's AC is less than 22, you always want to take the -5 to roll for +10 to damage (the average damage per round will be higher).
Your remaining two attribute increases should got to getting a 20 dexterity. So far we're up to: 5 attacks per round, expending nothing other than a 3rd level spell slot (2 per day available) and ammunition. Each attack does 1d6+15 damage, or 18.5 average damage per successful attack. That's (averaging out the rolls) a whopping 92.5 damage per round, without even using action surge! Even more amazingly, you could still be dealing an average 37 damage on a turn in which you are DODGING (haste action + bonus action attacks)!
Even without haste, we still could be doing an average of 74 damage per round (assuming all attacks hit) with the hand crossbow, expending no resources other than bolts.
For the final two levels, there are a couple of options; but probably the best one is to take two levels of Rogue. You'll get an extra 1d6 per round from sneak attack, and cunning action (which, although it would lose one weapon attack, would allow you to disengage, dash, or hide as a bonus action: it'll give you considerable versatility on the battle field, and allow you to easily keep your distance). Also, you could get expertise in stealth and acrobatics, which will be a huge boon if you're hoping to scout or hide in combat, or avoid being grappled. Our damage total (without action surge) is now up to 96 damage per round. If you wanted to Haste more often rather than do a bit more damage (and losing some battlefield mobility), you could take 2 levels in Wizard instead, allowing you 3 castings of haste automatically, and an Arcane Tradition (War Magic could fit nicely here, or Bladesinger if your DM allows it for a non-elven character. Warning: a DM may consider an attack with a hand crossbow to be using "two hands to attack with a weapon", which ends the bladesong. Check with them before choosing a subclass).
The Halfling's bonus to dexterity is essential, but you could get this from other races. A real benefit is the +1 bonus to constitution. It may not look like much, but this build is very much about maintaining the Haste, and has no attribute boosts to spare. If you take this build, DC 10 concentration saves (which is what most attacks will incur if they hit you; only attacks dealing 22 or more damage will up this difficulty) will only fail on a natural 1: and because of Halfling's "Lucky" trait, this means the standard concentration save will only fail 1 in 400 times. Also, the Halfling ability to move through the space of medium hostile creatures makes this build insanely mobile. If you went with 2 levels of Rogue, even if your character was completely surrounded by (medium sized) guards, you could Disengage (cunning action), Dash 45 feet (Haste dash action, speed doubled, five feet of difficult terrain for moving through an enemy's space), make three attacks (action), then move up to 50 feet more, all without incurring any opportunity attacks.
For your non-abjuration/evocation first level spell gained at level 3, I'd suggest Find Familiar (an Owl using the Help action in combat can be very useful). You should pick up Mage Armor right away at level 3. With mage armor and Haste activated, you'll have an AC of 20 (13+5 from dex+2 from Haste), before you even take cover (which you should be doing as much as possible as a ranged character). This will make your standard AC between 22 and 25. If you pick up Shield at level 5, you can even bring this up to 30 for a round, in REACTION to being hit (which could cancel the hit in question) at the expense of a first level spell slot.
There are other options that make for powerful ranged damage dealers at this level (Level 10 bard with swift quiver and heavy crossbows will fire 4 times per round, and ranger5/rogue* beats out fighter at most levels). But this build has a slight (or major) numerical advantage over all of them I've been able to calculate at precisely level 15. One last consideration: if you went Fighter 14, Rogue 1, you'd get the same max damage, lose the cunning action, and get access to one 3rd, 2nd, or 1st level Wizard spell of your choice. Mirror image alone might be worth this: but it all depends on how much flexibility you want on your turn. Personally, I'd go Rogue 2 (cunning action is fun and useful).
There are no WotC-released rules for this that you aren't already (per your Question) aware of
I've gone through all of the Unearthed Arcana and all the rulebooks and no such "get more cantrips instead of spells" variant rule exists.
The ways released by Wizards of the Coast to acquire more cantrips than your class and class features would naturally give you are Multiclassing, Feats, or starting play as a race that gives you a Cantrip such as a High Elf or Tiefling.
There are variant rules to allow you to cast more low level spells (see DMG288), but those aren't what you are looking for.
Homebrew Territory
As DM, it is entirely in your wheelhouse to invent rules to allow players to 'learn' new Cantrips. There are a few different ways I have seen this conducted. Unfortunately, I have not played with any of these rulesets, so I can't speak to how well they work in-game. But, hopefully, this will give you some ideas if you want to take the plunge into homebrewing up rules.
(Disclaimer: I don't remember where I saw each of these first, so I can't give you the original source)
Spell Preparing Casters can Prepare Cantrips
The gist of this rule is that it expands the rules for preparing spells to include Cantrips. Thus, for Clerics and Druids, they may prepare up to their 'known' number of cantrips each day, choosing any cantrips from their spell list. The Wizard works much the same way, but must have these Cantrips in their spellbook (they 'add' a new cantrip automatically any time their 'known cantrips' count increases--and can add new cantrips to their spellbook from spellbooks and scrolls as if they were 1st level spells).
Downtime to Learn Cantrips
Using the 'Research' (or referencing the 'creating a magic item' rules if you want it to be harder) Downtime rules from UA and, now, Xanathar's Guide I have seen homebrew rules that allow any caster who can cast Cantrips to use Downtime to learn a new Cantrip.
You can learn extra Cantrips as 1st-level spells
This rule essentially means that any Cantrip that you don't know as a Cantrip can be learned and treated like a 1st level spell. It isn't a spell you can cast by rote at this point, and thus consumes a Spell Slot, but it does give you access to Cantrips you wouldn't have otherwise. Paired with the Spell Points option on DMG288, this could let your players cast them a LOT as they get higher level.
I've seen a small variation on this where it suggests that every time a player casts a Cantrip as a 1st-level spell, the DM should roll a d100 or something...and on a 00, the player has 'mastered' the cantrip, and adds it to their list of Known Cantrips
Your own Suggestion...
If you want to permit a player to 'give up' a learned spell in exchange for another cantrip, then go for it (i.e. a Sorcerer should learn a new spell, but you allow them to learn a new cantrip instead). This is actually rendering the character a bit underpowered because spells are, naturally, more potent than Cantrips.
Summary
As mentioned, there are no WotC-released rules allowing you to gain more cantrips outside of Multiclassing, Race Selection, Feats, or the Cantrips (and at-will Warlock Evocations) granted by your class and class features. But, if you want to take the jump into Homebrew-land...hopefully these ideas I've seen around the net will help.
Best Answer
Here are the spells with healing powers
All spells listed below are available in the basic rules unless otherwise marked.
This search on D&DBeyond produces many (but not all) of these spells.