Here's my recommendation -- book three separate tickets: one from your mainland city to HKG, a separate ticket from HKG to TPE, and a final refundable HKG to USA ticket (or where ever her permanent residency is for).

When checking in for your mainland-HKG flight, show the refundable HKG-USA ticket and your USA green card to mainland exit immigration.

Once you arrive in HKG, cancel the refundable HKG-USA ticket and proceed to check-in for HKG-TPE as normal.