The PT-7, which we keep talking about using for a revised Crack but haven't done yet, does have a CT heater winding.
6080 heatar-cathode voltage rating is +/-300vDC. (Note the much-admired 5998 is only rated +/-100v.) Raining the heater voltage will of course reduce the heater-cathode voltage on the 6080, but in most cases the 12AU7 might be expected to dominate.
The issue with raising the heater voltage relative to the cathode is that is has two competing effects:
Raising the 12AU7 heater voltage will saturate the trace emission of the inside of the cathode sleeve to the (positive) heater, increasing its effective resistance and thus limiting the AC current that can flow between them. 20 to 50 volts is the usual amount. Note that you get the same effect by reducing the impedance from cathode to ground - Crack used an LED which has very low dynamic resistance.
But the heater winding is capacitively coupled to the adjacent high-voltage windings. In order to prevent this from causing even more hum, the raised voltage must be grounded at AC through a large capacitor. If this cap is too small, or has too high an equivalent series resistance, then you wind up introducing more noise than you are killing.