I'd say her power doesn't apply in that situation.
When encountering the Enemy Ship henchman, you have to follow the Encountering a Card steps, in order, and you can only use character/card abilities that apply specifically to that step.
The first step is to apply any "When you encounter" effects of the card. The ability in question doesn't specifically say you can use it at this time, and by the rules, that means it can't be used then.
Next is the Evasion step. Again, the ability in question doesn't say you can use it to evade.
Next is to apply any "Before you act" effects of the card. Again, the ability doesn't say it's usable at this time, so it's not.
Next would be the "Attempt the Check" step, and this is normally when you'd be able to use the ability to defeat a barrier.
However, as the Enemy Ship card doesn't have a check to defeat, you have to use this part of the rules: "If a card’s check section says “None,” look at the card’s powers
and immediately do whatever it says there."
The 'immediately' doesn't leave room for any use of abilities, as it happens at that instant.
I would say no. This is the same situation as using the Rum Bottle to defeat Pirate Hunting: you can't. This is because when you summon the ship, you are immediately starting a new encounter. Whether or not you defeat the summoned ship directly informs what happens with the barrier / henchmen.
The FAQ update specifically applies to doing things between the steps of the turn, not the steps of an encounter. See this part of the rulebook:
Encountering a Card During each of these steps, you and the other players may perform only the specified actions. Players may only play cards or use powers that relate to each step (or relate to cards played or powers used in that step)...Players may not play any cards or activate any powers between those steps.
So, you can play cards and use powers with out limits between the steps of the turn, but once you have an encounter you can only play cards that relate to the step you are on. And with a check to defeat of "NONE" Lirianne's power doesn't relate to any of the steps of the encounter with the Enemy Ship henchman.
As a Musketeer, Lirianne can take a power feat for the ability to discard a weapon in order to defeat a barrier with the Cache, Lock, or Skirmish trait. So, what happens when Lirianne encounters the Enemy Ship henchman? It is a barrier with the Skirmish trait. Similar powers state that it allows you to succeed at a check to defeat the thing it's letting you defeat, in which case this would not be a question, as there is no check to defeat the Enemy Ship, but Lirianne's power skips that phrase and simply allows you to defeat.
Therefore, where you would summon a random ship and make a check against the ship, reading Lirianne's power it seems I could just discard a gun and defeat the barrier without even bothering to see what the ship was (forgoing plunder and the ability to seize anything). Thoughts?
You still couldn't. When-encounter effects are faster than anything you can do; so before you can think of shooting the barrier, you are forced to encounter the summoned ship.