Isolated vs non-isolated dc/dc in multi-rail design
Either of your ideas will work.  With the already designed the block diagram.  the only thing that needs a change is: isolation is not needed because of the low voltages. No Isolation would also simplify the design.

"Correction would be done via a microcontroller."  Eeven an 8 bit micro 12/256 = only 46mV granularity for the 12 Volts and 20mV for the 5 Volts. These are fixed voltages, the outputs they should be stable.  If the design uses PWM then you might need to take control and ripple into consideration. Accuracy 

One transformer is going to be bulky and having separate channels means several transformers (if isolation is needed) there will be a transformer as well as the output inductor for each output. Or two bulky magnetics containing all the windings one in the transformer and one in the inductor.   

For voltage accuracy, there will be voltage drops in the wire resistance as well as in the output diode.  Changes in output current will result in voltage changes across the output diodes.  Having one bulky transformer will also mean only one supply being controlled and the rest just following it. 

Separate outputs will have accurate voltages for each output.  However, this is dependent upon your specifications. As for efficiency, it is usually best to calculate this in excel. It may be that the separate outputs will give a higher efficiency.