No. It isn't possible to have two readers connected. The ICs send 100 uA and 10 uA pulses to the diode. You would have to cut the trace at the pins (S-7 and U-7) so that the winbond circuit is no longer connected to the diode. This would I suppose kill any temp protection of the board, but you would get diode temps then
I wonder if there would be some way to just break the connection in the socket and run the wires from the maxim reader to the chip's pins like that?