I'm 99% sure this won't work, but...

[digiman619]

If I take War Soul or Gifted Blade, I trade away my psychic strike feature to get initiating or manifesting, respectively. If I later get the Psychic Strike blade skill, can I trade it away AGAIN to get the other archetype? As the title says, I'm almost sure you can't. You can get both in a "high psionic" game (which gets me GB in addition to Psychic Strike, which you then trade for War Soul, though, right?

[Forrestfire]

No, you cannot. Even if you could trade the ability itself, archetypes are chosen when you first take the class.

[Feint's End]

If I take War Soul or Gifted Blade, I trade away my psychic strike feature to get initiating or manifesting, respectively. If I later get the Psychic Strike blade skill, can I trade it away AGAIN to get the other archetype? As the title says, I'm almost sure you can't. You can get both in a "high psionic" game (which gets me GB in addition to Psychic Strike, which you then trade for War Soul, though, right?

Your guess was correct. You can't do that. You never actually regain the psychic strike class feature of the soulknife which is the thing you trade away.

Not sure about the high psionics game thingy. This might be more of a suggestion hidden somewhere in the book than an actual rule so it comes down to your dm.

If you have to choose one of the 2 for a character you are making pick gb for versatility and war soul for straight combat power (although gifted blade helps here is well especially if you are strengthbased -> expansion).

Another build path I personally suggest on every soulknife is soulknife 3, contemplative wilder 3, dark tempest 10, soulknife 3, contemplative wilder 1. Use fighters blade to get your blade up to 20 (although realistically you only need 19 ... also a bladeskill with 6 levels of soulknife though). On top of being an almost fully fledged soulknife (except for some bladeskills mostly) you get manifesting as a 14th level wilder (7th level powers you can choose anew every day).
Pick up powers you want via expanded knowledge (battle transformation is a good one).
With focused offense you can make wisdom your main stat and have really good safe dcs while still being a melee monster.

It's basically a much stronger soulknife (or gifted blade for that matter).

[ATHATH]

Could you take another class that grants Psychic Strike first, then trade away both of your copies of PS when you take your first level of Soulknife?

[Forrestfire]

No, because the archetypes can only swap abilities from the class.

Notably, even if you had multiple Psychic Strike abilities to trade, no soulknife archetype that replaces it would ever be compatible with another outside of a special note, because they still replace the same ability.

[grarrrg]

archetypes are chosen when you first take the class.

Almost, but not quite.
An archetype is basically chosen when the first replacement would happen.

It's buried in the retraining (http://www.d20pfsrd.com/basics-ability-scores/more-character-options/retraining) rules:
"Note that you don't have to use the retraining rules to take an archetype if your class level is low enough that the archetype doesn't modify any of your current class abilities. For example, if you're a 1st-level fighter who wants the archer archetype, that archetype doesn't replace any class abilities until fighter level 2, so you don't need to use the retraining rules at all—once you reach 2nd level, you can just decide to take the archer archetype."

So you could go to Bloodrager 19 without being a Primalist, and then suddenly decide 'you know what, I don't like this power, what else you got?'.

It still won't help him "double dip" though.