With your Rega DAC, I would stick with 24/96 upsampling. This should improve SQ. Many DACs actually sound better with 24/96 because their digital filters for 192 are not as good and the difference is very small if audible at all. Your Rega DAC is an older design. Here is what I read on Stereophile review:
"When you connect to your computer USB-to-USB, the signal goes to the Rega DAC's Burr-Brown PCM2707 USB receiver—the same chip found in Musical Fidelity's M1 DAC. This receiver chip is why you're limited to data rates of 48kHz and below and 16 bits via USB; but not so via the four S/PDIF inputs."
So I believe you need to use S/PDIF, not USB. You can improve things even more by using an upsampling reclocker like the Synchro-Mesh in the S/PDIF cable. This will get you lower jitter as well as upsampled data. The upsampling chip is critically important as is the master clock used. The $599 Synchro-Mesh delivers signal with ~20psec of jitter. It is equally important to use a really good S/PDIF coax cable, like the $499 Reference BNC (with RCA adapters) or the $275 Standard BNC cable.
Here are some jitter plots with different cables using the Synchro-Mesh:
So, how to get a S/PDIF signal from your laptop? The best way is to get an inexpensive USB to S/PDIF converter like a used Off-Ramp 3. You only need to support 16/44.1 for streaming an the Synchro-Mesh will upsample to 24/96 and give you low-jitter.