As a preliminary note, a good solution to this problem will (as usual) involve talking to your players. Ask them if they think this is even a problem, and if so, what kind. You may find that they enjoy being reckless and don't mind dealing with the consequences, even if that means inaction or even death for their characters. If so, congratulations: you're all set! But if they don't want to be unconscious and bored, read on.
You can try different things to keep them entertained, like having them run NPCs, narrate the combat to make it more epic, or take notes for posterity. If your players are flexible, that may work, and they may even enjoy the change of pace. (Though honestly, in my experience once they're stabilized it's usually a good time for a bio break to get more drinks for everyone or take care of other necessities, which should resolve the situation nicely unless it's a really long fight.)
However, most players are at the table because they want to play their characters. (Not all of them, but most of the standard motivations require being able to influence the game in some capacity, and most of those involve investment in a specific character.) If that's true, then no substitute will ultimately be as satisfying as being in there with your own character, mixing it up.
So if I could back up a bit, I would ask a different question, namely, why is this situation occurring so frequently that you have to ask?
If PC's routinely get KO'd, I suspect one of three things is true: A) your PC's are low level; B) your PC's are employing bad strategies; or C) your encounters are structured in a way that causes a lot of KO's, possibly because you're using Glass Cannons (warning: TVTropes).
A) should resolve itself pretty quickly. B) might resolve itself as your players learn better strategies, but you may need to ask if they want any pointers. If it's C), though, you can fix this by adjusting your tactics, both in building encounters and how you run them. A plan failing because "You didn't even dent her armor!" can be as spooky as one failing because they spotted you and stabbed you. (Of course, if you go too far with defensive enemies, fights can drag on forever, but sometimes people have to be defeated in a particular way that requires a little in-game research or experimentation.)
More importantly: if your boss fights are deadly and/or you've successfully cultivated an atmosphere where they feel deadly, when a PC does drop your players should be scrambling to get them back up so they don't lose the advantage of numbers (not to mention, y'know, keeping them alive). You've said the party doesn't have a ton of healing, but you can influence that by pointing them towards Goodberry and other useful spells and abilities, adding an NPC, or just giving out healing potions as loot (something that's working pretty well for 2 groups I'm in with relatively little healing). If people are watching each other's backs, a PC shouldn't be down for more than a round or so.
If you take the above into consideration, I think you can probably settle into a rhythm where this isn't too much of a problem in the first place.
Most of your constraints are solved with the updated Mystic class
With the March 3rd Unearthed Arcana Mystic class, you can, among other features, focus on one psionic discipline to gain a passive benefit. The Adaptive Body discipline says the following:
While focused on this discipline, you don’t need to eat, breathe, or sleep. To gain the benefits of a long rest, you can
spend 8 hours engaged in light activity, rather than sleeping during any of it
This doesn't solve the problem of not needing to drink, but you can get this benefit as a first level Mystic, meaning you can get most of want you want very early. Additionally, unlike most of the other options below, the requirement for sleeping is removed entirely, instead of just worked around with something like trancing. Ultimately, a Monk 15/Mystic 1 gains Timeless Body (described in detail below), which will remove all requirements for eating, drinking, breathing, and sleeping. Additional options are provided below:
The new Sorcerous Origins are an option
The Sea Sorcery Origin in the Feburuary 6th Unearthed Arcana has the Water Soul feature at 18th level:
Starting at 18th level, your being is altered by the power of the sea. You gain the following benefits:
• You no longer need to eat, drink, or sleep...
Combine this with a wish (described more below), and problem solved. Though you will likely spend more time without this power than with it, and you don't grow into this condition steadily, like some other options.
If you wish to avoid playtest material, the following options are available with just the core rulebooks.
Petrification Solves all your problems:
Turning yourself to stone, or other such material, means that you don't need to sleep, eat, drink, or breathe. Though, it's basically like being dead. This can be useful if you have a very reliable way to remove the petrification and you need to entomb yourself for an extended time span. Like magical cryo-sleep.
Move to the Astral Plane:
The Astral plane is described on Pg. 46-7 of the DMG.
The Astral Plane is the realm of thought and dreams where visitors travel as disembodied souls...Creatures on the Astral plane don't age or suffer from hunger or thirst.
Since this plane is described in spiritual terms, a sea of thought rather than matter, it could be assumed that the need to sleep or breathe is also removed while you inhabit the plane. Though this is up to your DM, since the DMG says nothing about those two processes. Also, in previous editions, if you return from the astral plane to any non-timeless plane, the effects of time will accumulate retroactively upon entering the new plane. For example, returning to the Prime Material plane after ten Prime Material years in the Astral plane will cause you to age 10 years, and the hunger, thirst, and sleeplessness of 10 years all take effect instantaneously. Basically, you will die if you leave the Astral plane without taking some serious precautions. This is omitted from the 5e description however. Finally, you will likely have to actually shift yourself into the Astral Plane. Projection, as with the Astral Projection spell, will leave your body limp and helpless on whatever plane you originated from.
True Polymorph yourself into an intelligent undead or construct
True polymorph allows you to change yourself into another creature. There is no rule that you can't change yourself into an undead, which will give you undead nature. You could also change yourself into a construct. However, all your stats, including your mental stats, are replaced by the stats of whatever you choose, and its CR must be less than or equal to your level. Also, literally change into whatever undead/construct you decide. You will be subject to all of the limitations of that form. That is: don't turn yourself into something that cant speak or doesnt have arms to cast spells
Assuming the new form is suitable, you can polymorph yourself and concentrate for the full duration to make the change permanent.
Timeless Body solves hunger and thirst
A 15th level monk gains the timeless body feature. You no longer feel the crippling effects of old age, and no longer need to eat or drink as ki sustains you. You will still die of old age though, and you still need to breathe and sleep.
Elven Trancing is not sleep:
Elves can get the benefits of 8 hours of sleep by Trancing for 4 hours. The trance is just deep meditation; the Elf never loses consciousness. And so, elves technically do not need to sleep.
Wish that you no longer need to respirate to live
Assuming you can get your hands on a Wish you could wish that you no longer need to respirate. You will need to word this very carefully as to avoid any potential twisting of the words. And ultimately, the DM still has the authority to turn the wish down. You will also suffer the stress of a wish from not replicating a spell of the 8th level or lower.
Combining these gets you what you want
So, if you are an Elven monk and you find a way to cast a Wish spell (example, find a Luck Blade), you will be technically sleepless, and have no requirement for food, water, or respiration. All without being undead or a construct.
Additionally, gaining these benefits in this manner will keep them active at all times, not only while you are conscious.
Immortality Too?
If you make it to 20th level, you could potentially get the Boon of Immortality (DMG pg.232).
If combined with the options described above, you will also stop aging all together, become immune to all aging effects, and you will no longer die from old age. At which point, you will be effectively timeless, sleepless, and have no need to eat, drink, or breathe.
Best Answer
Just cut a hole in his cheek.
Puncture wounds are healed by regular healing spells, so just stab the tortle through the cheek and shove a goodberry in there. It won't be fun, but it works. Or...
Find a cleric who can cast Regenerate.
Once you have done that (and I highly recommend doing it in this order), use a knife and just cut the Tortle's lips off, thread and all. Regenerate will restore the character's missing body parts, lips included, after two minutes. Strictly speaking, you don't need a cleric: a druid or bard who can cast seventh level spells will also work.
Alternatives to finding someone who knows that sort of magic: find a Ring of Regeneration. This is probably a lot harder, but technically, you only need to borrow it for 1d6+1 days for the body part to regrow (here, the tortle's lips) at which point you can return it.