There is no reason to to be isolating the relay- the NPN transistor and the relay coil already provide two different isolations themselves.
Your problem is the shunt regulator you've built. It will lose regulation when current exceeds 68 mA, which is likely since you're powering a relay coil.
The capacitor will do nothing at all to prevent the shutdown in the long term. It will provide a few milliseconds of power for the relay coil and then the voltage will drop enough to kill the PIC.
The shunt reg draws a lot of power from your supply for no reason, too. It draws 68 mA, the max supply capability, all the time whether the circuit needs it or not. Just add a common 3-terminal regulator. It will draw maybe 1mA above what the circuit consumes, and can supply up to an amp or so if it's heatsinked. For the circuit you show, it probably doesn't need a heatsink.
You can also use a NPN transistor or preferrably an N-channel MOSFET to drive the motor directly, exactly the way the relay coil is being driven now. A bit simpler. The transistor needs to be appropriately rated for the motor's current and a flyback diode must be in parallel with the motor. In fact, your circuit already needs a flyback diode across the relay coil.
A small ceramic or tantalum (0.1 uF is fine) on the PIC's power pins, right at the PIC, is always recommended although most applications will work fine without it. It filters the small surges of power with each clock cycle.