The Dxxxx series docks use DisplayLink chips to interface with the system, and I'm going to guess that given the D1000's age, the DisplayLink chipset version it has doesn't support 5.1 audio even through the HDMI output. You didn't specify which XPS 13 generation you have, but if you have one with a USB-C/Thunderbolt 3 port, you might want to consider a dock model that doesn't rely on DisplayLink at all since it can come with some serious drawbacks that I've described in the post marked as the answer on this thread. Since you mention 5.1 audio, I'm guessing you're probably trying to either watch movies or play games, and those are two of the absolute worst possible use cases for DisplayLink technology. If you don't have USB-C/TB3 and DisplayLink is your only option, then a dock with newer chipset would likely solve this (there are third-party DisplayLink docks too, fyi), but you might still find that DisplayLink delivers an unacceptable experience for what you're trying to do.
Btw, I found this guide from DisplayLink about multi-channel audio, although again I don't know if it will apply to your chipset.