Analog line level volume = voltage. If your amps are receiving a line-level analog signal from the computer, then higher volume is necessarily higher voltage. However, the card has a maximum output voltage that corresponds to Digital Full Scale (0dB). JRiver's volume control works in the digital domain, so if you just add boost in JRiver, you'll probably drive your music into clipping. Your realtek can only output as much voltage as it can.
You quote a .3V threshold and a 3 mV threshold (which is .003V), and I'm not sure which one is the right one; It seems like most computer audio would trip the second one; the first one might be harder. Most consumer PC soundcards max out (peak volume) at 2VRMS, so .3Vs would be something like -8dBFS (pretty high digital volume required).
Realistically, you're probably better off getting a DAC/Soundcard with a "hotter" output or finding another way to trigger the amps.