Kenneth,

Is your shack below ground or without windows?
My shack is in the basement. I run 600 ohm open wire
from the Palstar Balanced Tuner (with balun in the input) to
feed thru insulators in a Plexiglas insert in the Window. Then
600 ohm to the doublet using string to the separators to
hold the feedline in a position away from the house.

The Palstar Tuner is similar to your description. It tunes my
210 foot doublet on most frequencies and its L equivalent
circuit is broad enough to make retuning for small frequency
changes unnecessary.

If you use your current balun on the input, an L (unbalanced) looks
like a balanced circuit on the output. Of course you must
insulate your capacitors above ground, ditto the inductor.

The ARRL Guide to Antenna Tuners has an excellent discussion
on balanced vs. unbalanced lines that you might wish to read
as well as balanced vs unbalanced tuners.

73,
Ed, N5EI