Update: I finally got back to this project and can narrow down what might have caused this issue for anyone else encountering it.
I may not have specified before but the chip in question was installed on a laptop motherboard and I was using a Ponoma clip.
I tried running flashrom with the same CH341 programmer and the same wiring harness numerous times and I always got the same results with inconsistent id's.
Per random internet advice, I tried with power and battery removed (for days, so any capacitors would have been fully discharged), with battery installed, with the charger plugged in, and every permutation I could come up with. Same results.
I was never super confident that the Pomona clip was making good contact but I can't believe it didn't make good contact at least one of those times.
I figured that either my CH341A was busted or I'd somehow got the wiring harness wrong, so I bought some new W25Q64FW chips just to test it out. Turns out my equipment works perfectly on these loose chips.
So I removed the original chip from the motherboard and found that my programmer works just fine on it now that it's free.
S long story short, the problem was that it was attached to the laptop. I assume that there is some interconnection between the legs within the motherboard which prevents the chip from being programmed. Whether this is intentional, incidental, or a flaw I do not know.