In Trespasser, your Inquisitor ends up losing the arm that bears the Anchor / Mark, whether they like it or not. This naturally means that their combat performance will be adversely affected.
They can adapt and adjust their combat style to fit their new set of circumstances.
So realistically, what kind of Inquisitor will be able to survive that ? By this I mean what class of Inquisitor will be able to survive that ?
Well, that's incredibly Ableist of you. Compare the loss of one limb to the loss of life (because a person loses all value the second they're not 100% able-bodied), and place all worth of the person in the thing they can no longer do. If you can't fight exactly as well as you used to back when you had that extra arm, then you can't fight well at all. That extra fighting ability that came with that extra arm is where ALL the combat talent and worthiness lies, so now that you don't have it anymore don't bother trying to adjust your combat style to fit your new circumstances or work around your disability. Just give up fighting completely, go home and sit down and wait for death.
A Two Handed Warrior Inquisitor cannot wield a Two Handed weapon anymore.
So? They can learn to wield a one-handed weapon.
A Sword and Shield Inquisitor can wield a sword on the functioning arm but they cannot wield a Shield anymore.
Again, going with the obvious: learn to fight with just the sword. However, I think this class is the most prosthetic-limb friendly since you don't necessarily need to replace functioning fingers, just a hunk of solid to attach the shield. They can attach a prosthetic forearm, attach a shield to that prosthetic forearm, and then retrain themselves to fight sword-and-shield with that new prosthetic forearm.
A Mage Inquisitor requires two hands to use a staff and perform the hand motions for necessary for spellcasting. Okay, maybe they can learn to use a staff with one hand but they still need another hand for spellcasting.
A mage only really needs one hand to hold a staff, and then to cast spells by pointing the damn staff and firing spells.
Okay, they can no longer do a bunch of fancy two-handed flips and twirls with their staff, but that's not exactly necessary for combat. They can, again, retrain themselves to fight one-handed.
Knight Enchanter Inquisitor is further screwed since they have to choose between empty hand for spellcasting or spirit blade or staff.
That's no different from how Knight Enchanters already function. Have you actually played as one? The Inquisitor already can't use the spirit blade and cast spells at the same time. That is, the game doesn't let you strike a foe with the spirit blade and shoot spells with the staff at the same time. This is not exactly a huge loss, here.
An Archer Rogue cannot use a bow anymore since bow requires two hands. Yes they could use a crossbow but they need another hand to reload the crossbow which they won't have.
One-armed archery exists.
Also, necessity is the mother of invention. They could probably get Dagna to invent a way to reload a bow / crossbow without that extra hand.
A Double Dagger Rogue is have to make do with using just one dagger, making them a Single Dagger Rogue but at least they can function as they used to with some reduction in damage.
Oh, that's funny. A two-handed warrior can't learn to fight with a one-handed sword, and a sword-and-shield warrior can't learn to fight without the shield, and a mage can't learn to just point and shoot with their staff one-handed, but ONLY a double-dagger rogue can learn to fight without an extra dagger. I see how it is.
Thoughts ?
As others have said, a simple prosthetic limb could take care of most of the problems you listed above. So could inventive new combat techniques and even weaponized inventions to adjust to and make up for the lack of that extra arm. Sure, fighting without that extra arm is more challenging, but it's not impossible. Sure, the Inquisitor will probably not fight as well as s/he used to for a long time, but that doesn't mean s/he cannot fight well at all. Sure, there will probably be a difficult adjusting period where the Inquisitor re-trains themselves how to fight without that arm and/or with a prosthetic limb, but that doesn't mean they can't fight at all, nor that they should just sit at home and lament all the things they can't do without their left arm and wait for death.