the 4:1 balun built into the tuner is for using 300-600 ohm ladder line / twin lead / open feed line
from what I have read you want to use the single wire terminal on the tuner and have your tuner grounded good,,, ,in that case you just run a single wire from the tuner to outside and where ever your going to run it, the entire wire becomes your antenna. the 4:1 balun is not used in that case.(remember there is always 2 halves to an antenna and the wire from the single output will be working against the tuner ground system) this set up usually doesn't work well but that's what radio is about, trying different things
if I were you I would use the balanced line output of the tuner with your choice of (300 ohm twin lead, 450 ohm ladder line or 600 ohm true ladder line open wire feeder) run that outside and make the longest dipole you can(even if you have to bend the ends around a corner) whatever you can get in the air, feed it direct with the twin lead/ ladder line or 600 ohm feed line
I have 2 hf mutli band wire antennas , 1 is a 80m doublet fed with 600 ohm true ladder line/open feed line to a Johnson match box and the other is a 40m dipole fed with 300 ohm twin lead to another Johnson matchbox or my mfj993 auto tuner. both work multi bands and no rf in the house at all,,, I cant believe how easy it is to work with really.