Argthtjtr

We already know from Sage Advice and other answers to other questions that by RAI, (and I guess RAW too), Cutting Words (and Bane) don't work on critical hits, though in the case of Cutting Words they can still reduce the damage by an amount.

The Grave Domain's 6th level feature, Sentinel at Death's Door states that:

As a reaction when you or a creature you can see within 30 feet of you suffers a critical hit, you can turn that hit into a normal hit. Any effects triggered by a critical hit are canceled.

Does that mean that, since it is no longer a critical hit, that Cutting Words can now apply to it properly?

A follow up question: If the above is true then if a Bard cast Cutting Words before the Cleric used Sentinel at Death's Door on a critical hit, does Cutting Words still apply?

Yes, these two features combined could potenially cause a critical hit to miss.

As quoted in your question, Sentinel at Death's Door, on it's own, turns a critical hit into a normal hit.

Cutting Words, crucially, can be used:

[...] after the creature makes its roll, but before the DM determines whether the attack roll or ability check succeeds or fails, or before the creature deals its damage. (emphasis mine)

Under normal circumstances an attack roll of 20 will always succeed:

If the d20 roll for an attack is a 20, the attack hits regardless of any modifiers or the target's AC. This is called a critical hit [...]

So, if a critical hit 'hits regardless of any modifiers or the target's AC' the fact that this is no longer a critical hit means it no longer 'hits regardless of any modifiers or the target's AC' and is now a viable target for cutting words.

Cutting words could not have been applied 'after the creature makes its roll, but before the DM determines whether the attack roll or ability check succeeds or fails' because as soon as a 20 was rolled it was classed as a critical hit and the DM did not need to determine whether it would succeed or fail - it was an automatic success.

However, once the Grave domain cleric has used their reaction to negate the critical hit there is a second opportunity to use Cutting Words 'before the creature deals its damage'. So, now the Lore Bard can use Cutting Words to try and cause the attack to miss.

I believe this is balanced as it has a (relatively) high opportunity cost.

The party will require both a Grave domain Cleric (level 6) and a Lore Bard (level 3) to have access to both of these features. Both abilities consume each player's reaction so a multiclass Cleric/Bard could not do this on their own. Sentinel at Death's Door can be used a limited number of times per long rest and Cutting Words spends one use of Bardic Inspiration die.

Cutting Words used this way also has a higher than average degree of failure. By which I mean that, statistically, the Bard will likely have to roll pretty high on their Bardic Inspiration die, in order to cancel a hit from a 20 on the dice, plus any modifiers the attacker adds, in order to bring the total below the AC of the attack's target.

A Bard who is playing the percentages might prefer to save their Bardic Inspiration to use when it is statistically most likely to cancel damage.