You should be able to use the balanced outputs and single ended outputs of your DAC at the same time. I did that with my DAC for a while, using a headphone amp's balanced inputs and a preamp's single ended inputs, and it worked fine. Did you try a balanced cable to your amp, and single ended cable to your headphone amp yet? Just curious.....
By the way, that particular switch from Markertek is for two sources and one destination. Two females XLRs in and one male XLR out. Make sure you get one with the right connectors.