Interesting. For whatever reason, despite members of my group dual-wielding everything from MIU Weapon interfaces to pistol & wrist-mounted pistol on the same hand combos, this has never actually come up.
Since I'm a stickler for RAW, let me quote you the following:
"• The character may use any melee weapons or ranged
weapons that can be reasonably used in one hand.
• The character may use either hand to make an attack."
So, on a purely technical level, I'd say this implies you are not allowed to use any weapon you're not holding in either hand, so the answer would be that you can't use your beak at all. If you wanna get super-anal, the power fist isn't *in* your hand either - though it is still your hand you're attacking with, so that could go either way. Next, the rules go on to say:
"Attacks made using the character’s secondary hand suffer
a –20 penalty to Weapon Skill or Ballistic Skill Tests."
So if you're not using your secondary hand, which is what this rule specifically refers to, you wouldn't take the -20 penalty. You would still take the -20 penalty to *both* attacks from using Two-Weapon Wielder in the first place, however, and you *do* need that talent in order to fight with two weapons in this manner.
Since the rules do leave a tiny bit of room for interpretation, let me say that I would allow it. Ambidextrous is not particularly devalued by this ruling (says I), and having to rely on crappy natural weapons or ridiculous-looking MIU interfaces to compensate is punishment enough.