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Legio Custodes have the Preternatural Skill special rule that gives them +1 Initiative in Assault if their Weapon Skill equals or exceeds that of their opponent. Would this rule apply even when a Custodian is using an Unwieldy weapon(such as a Solerite power gauntlet) which would make them strike at Initiative 1?

Depends on what the Unwieldy rule says. Does it say your Initiative actually is 1, or that you strike at I1?
No rulebook on hand where i am, but i suspect it's the latter, which initiative bonuses would not help with.

I've asked elsewhere, and its a basic rules fail on my behalf. Set values always trump modifiers(unless stated specifically in the rule). Unwieldy clearly sets to Initiative 1 and Preternatural Skill doesn't have any special exemptions in its wording.

Unweildy states you STRIKE at I1, so that would overide anything