No, Bardic Inspiration can't cause a critical hit

Cutting Words states, "...subtract the number rolled from the creature's roll.", so the number is being applied to actually change what you rolled on the d20.

Bardic Inspiration states, "add the number rolled to one ability check, attack roll, or saving throw", so the number is just another bonus on the attack roll.

The difference in phrasing is very small but important here. Terms like, "attack roll", "saving throw", "ability check", are common phrases in 5e used throughout the book. PHB, p.195, explains that you make an attack roll by rolling a d20 and adding appropriate modifiers. If it weren't for the Jeremy Crawford tweet it would be sensible to assume that the, "roll", in Cutting Words refers to the, "attack roll", but that tweet exists so instead it refers to the actual roll of the d20. Bardic Inspiration specifically says, "attack roll", so we can be relatively certain that it is just another modifier added to the roll.

The spell Bless uses the same phrasing as Bardic Inspiration, "...and add the number rolled to the attack roll or saving throw", and there is no reason to believe that it grants (or bizarrely prevents) critical hits.

The gist: your Armor Class remains the same despite being unconscious. You can justify a miss by saying the unconscious target was not hurt by the attacker's careless attack.

Yes, your AC is still kept the same

Your thief's AC is 15, before and after succumbing to the sleep spell. That is simply how AC is calculated (Base AC of Armor + Dex mod), as written in PHB 144.

PHB 144

Armor Class (AC). Armor protects its wearer from attacks. The armor (and shield) you wear determines your base Armor Class.

Followed by the AC specified in PHB 145 on how to calculate Armor Class from Padded or Leather armor: \$11 + \text{Dex modifier}\$

How is this narratively justified?

Armor Class is not how hard you are to hit, but how hard you are to wound.

Armor Class, PHB 14

Your Armor Class (AC) represents how well your character avoids being wounded in battle.

The difference is usually immaterial, but in this case, it is important. The Barbarian could still have hit the Thief, they just failed to hurt them.

We can also take guidance from how Armor Class is calculated for objects:

Statistics for Objects, DMG 246

Armor Class. An object's Armor Class is a measure of how difficult it is to deal damage to the object when striking it (because the object has no chance of dodging out of the way).

Objects cannot dodge, much like a sleeping creature, so you instead use AC to represent how hard it is to damage the object. Interestingly, the following ACs are used for the brittle materials:

\begin{array}{|c|c|} \hline \textbf{Substance} & \textbf{AC} \\ \hline \text{Cloth, paper, rope} & 11\\ \hline \text{Crystal, glass, ice} & 13\\ \hline \text{Wood, bone} & 15\\ \hline \end{array}

Your unconscious Thief, as it turns out, has the same AC as bone. This means, even after being hit by something (because it can't dodge), bone is still difficult to damage. It is probably resilient enough that you cannot crack or snap it in one blow.

In other words: you can let your Barbarian's swing hit the Thief, but you can justify by saying, despite the impact, did not actually hurt the target. They could bruise, or get a cut, or a small puncture wound, but the nature of the blow is such that the target was not meaningfully harmed.

Also narrate the failure of the Barbarian

Despite having Advantage, your Barbarian's best roll was still a 14. That is a failure on their end, not just a product of the resilience of your unconscious Thief. Focus on that as well.

You can say that the Barbarian thought they were aiming for the head, but they lost their footing slightly during the swing, or aimed too quickly or too excitedly, or the hilt on their weapon slacked just a bit. You can also add an external factor to the blow, and use that external factor to explain why they failed to deal damage.