Is this Circuit Valid?
The reason you'd fit a fuse is to protect the wiring.

You state that you'll be running this from a 3A 12v power source. If the source is current limited, and cannot deliver more than 3A, then as long as your wiring will handle 3A, so at least 0.3 \$mm^2\$, then there's no need for a fuse. 

If you substitute a car battery for your PSU, then you should fit a fuse.