Barbarian Unarmored AC10 + Dec + Con.
Thri-keen start with a 13 AC due to the Chitin.  Would this equation be modified to account for the +3 Unarmored Bonus(Chameleon Carapace) or is this negated.

No.

While you aren’t wearing armor, your carapace gives you a base Armor Class of 13 + your Dexterity modifier.

Your base of 13+ Dex is only applicable if ,you are not wearing any armor. When determining an AC, you can choose to use *either* the Barbarian unarmored, or take your base, because the RAW say you take the one that is higher (thri-kreen).

That said, the only person who actually matter is your DM. If they say it should be 13 plus Dex plus Con, then fine. What anyone else says, including me, doesn't matter and shouldn't be used to argue otherwise with a DM.

The rule is that you don't mix AC formulas.  Chameleon Carapace as 13 + DEX mod is a separate formula from Unarmored Defense's10 + DEX mod + CON mod, so you can't add the extra 3 to Unarmored Defense and wind up with 25 AC while wielding a shield.  Instead you are welcome to use whichever AC calculation gives you the highest value.

The AC mixing rule is in Sage Advice, here:

How do you calculate a creature’s Armor Class (AC)?

Chapter 1 of the Player’s Handbook describes how to determine AC, yet AC calculations generate questions frequently. That fact isn’t too surprising, given the number of ways the game gives you to change your AC! Here are some ways to calculate your base AC:

Unarmored: 10 + your Dexterity modifier.

Armored: Use the AC entry for the armor you’re wearing (see PHB, "Armor and Shields"). For example, in leather armor, you calculate your AC as 11 + your Dexterity modifier, and in chain mail, your AC is simply 16.

Unarmored Defense (Barbarian): 10 + your Dexterity modifier + your Constitution modifier.

Unarmored Defense (Monk): 10 + your Dexterity modifier + your Wisdom modifier.

Draconic Resilience (Sorcerer): 13 + your Dexterity modifier.

Natural Armor: 10 + your Dexterity modifier + your natural armor bonus. This is a calculation method typically used only by monsters and NPCs, although it is also relevant to a druid or another character who assumes a form that has natural armor.

These methods—along with any others that give you a formula for calculating your AC—are mutually exclusive; you can benefit from only one at a time. If you have access to more than one, you pick which one to use. For example, if you’re a sorcerer/monk, you can use either Unarmored Defense or Draconic Resilience, not both. Similarly, a druid/ barbarian who transforms into a beast form that has natural armor can use either the beast’s natural armor or Unarmored Defense (you aren’t considered to be wearing armor when you use natural armor).

The purple line is the important bit - different means of calculating your AC are mutually exclusive, so having more than one means you have to pick the one you want to use.

So the best armor you would wind up with would be 13 + Dex until your Con reached a +4, at which time it would be the new way of calculating your AC.  It does take a little of the MAD pressure off the barbarian, as you start with the equivalent of an 16 Con (+3) when calculating your AC.  Of course as a Barbarian, you want a high Con anyway, but this would allow you to boost Strength earlier if you preferred.