Daniel, check the voltage at the junction of C1, R3, and the gate of Q1 and Q3 when the PTT is active. That should be near zero. Q3 is the receive audio gate and should be open when transmitting. If there is a voltage more than 1.5 volts (or so) then that gate will remain open and you will get a click. Find the reason that it is high.
If that voltage is low. Check to make sure that the ground connection is good by measuring from the mod boards ground to ground on the main uBITX.
If the ground is good, your solder connections all look good, and there is a low voltage present at the first test then replace Q3. That will be a first, those little MOSFETs are tough. But anything can fail. Keep us posted, please.
Don, ND6T