Rules support for a Rule of Cool attempt

You are in the territory of a GM ruling, and since called shots and attacks on specific body parts aren't covered in the rules, this will probably break down into a three step process (at least) if your innovative PC wants to make this work. It won't be easy, but why not let them try?

1. Ability Check: Nature or Arcana

   The player needs to do the research to find out the weak points of a Tarrasque, and if it in fact has lungs. Let's say say that they pass that ability check, and they find out that Tarrasque does indeed have lungs¹. (DM ruling; nothing says yes or no in the monster description). So let's roll with this, or you just stop here if the ruling is "no lungs."

2. Ability Check, Opposed Athletics

   Now the PC wants to get into the throat, and then expand into Huge size (see below) without getting swallowed by the Tarrasque.

   • A non-trivial technical limitation here is that a Tarrasque can swallow a Large creature, so your PC needs to be able to turn into a Huge Creature, like a T Rex or a Stone Giant. This may call for True Polymorph, rather than Enlarge or Reduce; or, perhaps Polymorph into a Huge creature and be able to "reduce" (per the spell) to become Large to get in, and the dispel / stop being reduced to turn Huge. Unless your PC is already Large, getting from Large to Huge will take some work. Wild Shape? Shapechange? True Polymorph? Note that the PC can't stack enlarge spells to make this work.

   The effects of the same spell cast multiple times don’t combine, however. Instead, the most potent effect—such as the highest bonus—from those castings applies while their durations overlap. (Basic Rules, p. 81; Combining Magical Effects).

   Assign this research project to your PC: most players enjoy a challenge. Research Topic: How to get Huge inside a Tarrasque's Throat?
   
   
   Once inside the Tarrasque's head/throat, the difficulty is in getting to the "just right" spot in order to expand and try to collapse some lungs, or just choke the beast, since the Tarrasque has this particular ability:

   Swallow. The Tarrasque makes one bite attack against a Large or smaller creature it is grappling. If the attack hits, the target takes the bite's damage, the target is swallowed, and the grapple ends. While swallowed, the creature is blinded and restrained, it has total cover against attacks and other effects outside the Tarrasque, and it takes 56 (16d6) acid damage at the start of each of the Tarrasque's turns.

   Since the character isn't yet grappled by the bite, per the bite attack description (MM, Tarrasque) the issue of being swallowed is in doubt. As DM, this is a fine time to call for an opposed Athletics check (see rules on Contests) with the character's Athletics ability versus the Tarrasque's(Strength) check. The Tarrasque's +10 modifier ups the challenge here. There is a chance to succeed, or a chance to fail. Feel the tension building! If the PC fails there is a chance for bite, swallow, and more as consequences of leaping into the Tarrasque's maw.

   Consider whether or not Advantage or Disadvantage applies here. The DM can always rule that circumstantial advantage, or disadvantage, applies. (Basic Rules, p. 4)

   Granted, you as DM could assign a high DC (25-30, maybe higher) for this very difficult task without creating an opposed check. (DMG, DC's; PHB; DC's).

   If the PC manages to pull this off, there is good reason that they are able to remain stuck in the Tarrasque since the thing trapped inside of it is larger than a Large creature; it won't be barfed up by doing damage to it internally per the "barf 'em up" feature of the Tarrasque:

   If the Tarrasque takes 60 damage or more on a single turn from a creature inside it, the Tarrasque must succeed on a DC 20 Constitution saving throw at the end of that turn or regurgitate all swallowed creatures, which fall prone in a space within 10 feet of the Tarrasque. If the Tarrasque dies, a swallowed creature is no longer restrained by it and can escape from the corpse by using 30 feet of movement, exiting prone.

   OK, the PC has collapsed the lungs, or cut off the airway, now what? The PC is likely restrained, and may wish to remain so for a while. Why?

It will probably take a while for the Tarrasque to die.

Suffocating
A creature can hold its breath for a number of minutes equal to 1 + its Constitution modifier (minimum of 30 seconds). When a creature runs out of breath, it can survive for a number of rounds equal to its Constitution modifier (minimum 1 round). At the start of its next turn, it drops to 0 hit points and is dying, and it can’t regain hit points or be stabilized until it can breathe again. Basic Rules, p. 65)

The Tarrasque has a Constitution of 30; +10 ability modifier. It can hold its breath for 11 minutes, and then when it is out of breath it can choke to death in 10 rounds. (Another minute). You, the DM, need to decide whether or not the PC also has to deal with a similar problem while stuck there in the Tarrasque's throat?
Is oxygen starvation on the table for our brave adventurer? There are some magic items and spells, and some class abilities, that might mitigate this ... Necklace of Adaptation for example.
The DM can also adapt the suffocation rule a bit, as needed, to help with making this play out more smoothly.

If this whole thing succeeds, the PC can wear his, or her, "I Made the Tarrasque Gag!" t-shirt with pride.

This isn't a case of an "I Win" button; this is more like a mini-adventure within the larger "fight the Tarrasque" adventure, and it isn't a sure thing. That's a good thing: what reward if there's no risk?

¹ As @MarkTO noted in this comment: _{Many creatures do not have a combined airway/esophagus. This is how they can drink endlessly without having to pause to breathe like humans. Therefore, climbing into its mouth and enlarging might make it gag but wouldn't cut off its airway. That makes this a part of the aforementioned research project. See this Zoology stack exchange question for details (warning, graphic imagery).}