Is the whole chain
high impedance unbalanced input ->
DI box ->
split the balanced low impedance output -> one way to the board, one way to this monitor thing
?
and the monitor thing is an unbalanced input?
The easiest way to unbalance a signal, and one which almost always works fine, is just shorting pin 3 to pin 1. This is what happens when you plug a TS connector in to a TRS jack, or in the diagram posted above.
The problem with doing that is this case is that by unbalancing the connection to the monitor do-hicky thing you are also unbalancing the connection to the board. You could leave pin 1 floating at the unbalanced end of the connection. If you do this leave it connected at the balanced side, and I don't think you want to lift the ground. That's your only shield, and it's only connected at one end.
Any reason you can't use the parallel output on the unbalanced side of the DI box to feed the monitor?