If you are running the ladderline to a tuner, then
adding a ground strap and another coax shouldn’t
be a major problem, especially if you can keep them
against the side of the tube equidistant from the two
wires of the twinlead. Yes, there will be a small change
in the line characteristics, but the tuner will take
care of it.
That’s not to say that it is the ideal solution, but you
should be able to get it to work.
If the twinlead is truly balanced (not always the case
with the baluns provided in many tuners) and the added
wires are properly positioned with respect to it inside
the tube, there will be little induced current on the added
wires. In practice there may be some. If this is a
problem, then adding a choke/balun in the coax may help.
Details will depend on the specifics, and the purpose of
the ground wire.