Only the Star Spawn Hulk is at risk for becoming Incapacitated

As you quoted above, the only relevant aspect of Psychic Mirror is:

If the hulk takes psychic damage, each creature within 10 feet of the hulk takes that damage instead

The Psychic Damage is displaced, but the DC 19 Constitution Saving throw is not something that is off-put. There is no language there that says the Hulk avoids the risk of the Incapacitated condition. In addition, the Star Spawn Hulk is not immune to the Incapacitated condition.

In addition, there is no language that suggests those suffering from the redirected psychic damage have to make an additional save against Incapacitated. They only end up taking the damage.

Preventing Comet Staff abuse

One possible reason for the language not including either protection for the Hulk against the save or forcing the save on any creature within 10' of the Hulk is that it prevents a a Seer-Hulk combo from occurring. Normally, the Seer only targets a single creature with the Comet Staff attack. If this combo was allowed, it could then target multiple creatures for damage and condition. That's a HUGE difference in power for that attack.

On the flipside, it seems like it would have been a cool natural combo if allowed. If the Hulk was immune to the Incapacitated Condition and that condition save was passed onto the 10' radius targets, then this mechanic would be incredibly powerful and a neat effect.

Option 1: Once Damaged, Twice Healed

Assuming Life Transference is valid for use with Twinned spell (see below), the key phrasing that leads to this outcome is the following from the Twinned Spell description:

...you can spend a number of sorcery points equal to the spell's level to target a second creature in range with the same spell.

This specifies that we don't cast the spell twice, but rather the effect of the same spell applies to two targets instead of one.

The relevant part of Life Transference is:

You take 4d8 necrotic damage, and one creature of your choice that you can see within range regains a number of hit points equal to twice the necrotic damage you take.

The amount the target gains is equal to what you lost. Splitting wouldn't meet the effect of having each target gain an amount equal to what you lost. It's worth noting that spells that deal damage aren't split either, so it shouldn't for healing.

However...

There's an argument to be made that Life Transference targets both the creature and yourself, as you pointed out yourself in option 3. Under this interpretation, Life Transference would not be a valid spell for use with Twinned Spell.