You can convert it to use coax with power up the coax, and still have reasonable results. Simply use a 100nF capacitor in series with L6 to isolate the DC feed from the output, and connect the 13V supply wire to the centre core instead. You still need the balun on power, but connect the -13V5 side to the shield instead.
Bottom end you will need to feed in the power using another balun, basically replicating the input side in reverse, and feeding in your 12-15VDC supply to the new inductor, and connecting the ground lead to the shield. 100nF capacitor to isolate the DC from the antenna input and you are done.