No.
The 12-volt CT winding (6-0-6v) can deliver about 0.5 amps RMS current; the 12AT7 consumes 0.15 amps so there is 0.35 amps to spare. That's about 4.4 volt-amps.
The center tap must be grounded since this winding serves as an electrostatic shield for the 2.5v windings. Are you thinking perhaps of generating a +/-12vDC supply for some other circuit? If so, that could possibly be done.