Help me decide if this opamp circuit will work as purposed
No, your circuit will not work. The main problem is that 12V is connected directly to Vout. And Q4 is going to short 12V to ground (which will destroy Q4).  

Look at this schematic. Just for inspiration.







simulate this circuit &ndash; Schematic created using CircuitLab

UPDATE:
I changed value of R2 from 120kΩ to 90kΩ. It is wrong in question too. If OP wants to amplify from 0-3V to 0-12V than there should be 1:3 ratio for R1:R2

UPDATE2 - EXPLANATION:
This circuit makes Vout = 4* Vin  

If you increase Vin, output of U3 will go lower, which will make mosfet M1 more open (Vgs is more negative = Rds is lower) and thus Vout increases. It will increase until voltage at non-inverting input of U3 is equal to inverting input (which is Vin).  

If you increase Vin, output of U3 will go higher, which will make mosfet M1 less open (Vgs is less negative = Rds increases) and thus Vout decreases. It will decrease until voltage at non-inverting input of U3 is equal to inverting input (which is Vin).