Are you trying to mechanically toggle the fireplace switch? That’s going to be a though and probably unreliable. Can you mount a low voltage relay (like 12VDC) next to the fireplace switch (as you have pointed out the thermocouple only puts out a few millivolts) and use a WiFi switch powering a 12V DC power adaptor to energize the relay? Just a thought.
Disconnect bell wire wire from transformer and measure voltage at the screw terminals. Should be 24VAC. If nothing, internal fuse link in transformer is open. Transformer is toast.