Biggs' ability doesn't switch from off to on when I move from picking a weapon to picking a target; it's there all the time.
What makes you say that?
That's counter to every way a rule book or order of processing in any flow system I've worked with has ever worked. You cannot target someone unless you are at a stage where you are picking a target. You cannot Target Biggs because the weapon you've selected doesn't allow it so his ability no longer effects your ability to target other ships at range 1 to him.
Bigg's ability is an "Always On" condition. While Biggs in on the board, ships within range 1 of him are not valid targets for ANY weapon.
This really doesn't matter. Does anyone doubt that there will be a new FAQ VERY soon (I would be shocked if it wasn't tomorrow) and does anyone really think that Biggs won't still work the way he is supposed to?