Using 12V DC to create symmetric power source
You would do much better by using an opamp to make the ground point.

I sugget using a circuit like this:



Just use your 12 V instead of the 9V battery.
It depends on the opamp how much current can flow through the ground connection.

This circuit is also much more efficient than your proposal.

To make it suitable for higher (ground) currents, use a power-opamp (some audio power amps are also suitable) or see WhatRoughBeats's answer for a discrete solution. Note that that solution has no over current / over temperature protection.