Use a OR wired with diodes to control the gate of a PMOSFET
You can use something like the below. The comments indicate your problem- the gate must be allowed to rise to close to the +12V source voltage in order to completely turn the MOSFET off. This is done in the below circuit with an open-collector RTL NAND gate, with an inverter on one input, which I think preserves the logic you are trying to implement. 

You could replace Q1-Q3 with 2N7000 or 2N7002 MOSFETs and eliminate R3 &amp; R4, making it a bit simpler. 







simulate this circuit &ndash; Schematic created using CircuitLab