I dunno... I'm pretty certain that any FO with the testicular fortitude required to actually be sitting in the seat shouldn't have a problem telling the captain that while the left half of the cockpit may be accepting the extension, the right half will be too fatigued to continue.
That's fair enough and what I'd do as well.
But FYI, here is what it states:
117.19 Flight duty period extensions.
(a) For augmented and unaugmented operations, if unforeseen
operational circumstances arise prior to takeoff:
(1) The pilot in command and the certificate holder may extend
the maximum flight duty period permitted in Tables B or C of this part
up to 2 hours. The pilot in command and the certificate holder may also
extend the maximum combined flight duty period and reserve availability
period limits specified in § 117.21(c)(3) and (4) of this part up to 2 hours.
And since that's the way it's written, I would imagine most airline schedulers are taught to deal with the PIC about extensions. ATN saying he'll call in and refuse an extension isn't really addressed here. The FO at this point can call in fatigued which is fine, but the way the rule is written, it's not up to him to tell the scheduler that he won't accept an extension. Fatigue and extension are two different things. The way the rule is written, the SIC would have to tell the PIC about him not wanting to extend, and the PIC would have to make that final call.
As for me personally, I don't fly fatigued and I have never had to be extended because the route structure is mostly one leg transcons. As long as I'm not tired/fatigued, I would accept an extension by letting the PIC know, and let him/her talk with the scheduler. That's how I take the FAR 117 rule.