I'd say it's not worth spending more money to get more performance out of this or any other 7.1 headset. Just use it in stereo mode if you have no 7.1 source.
Operating systems and applications like games or media players are responsible for handling sound: selecting the appropriate output path and channel profiles, applying dynamic range compression where ...
Have you heard about O2+ODAC by NwAvGuy? After some evaluation of products in the market at around 2012 he started his own DIY project for a headphone amplifier (the O2) and later retrofitted a DAC into the design (ODAC, both together make up O2+ODAC). It's a USB Audio Class I device and therefore runs on a lot of platforms out of the box that support USB ...
The Asus Xonar D2X appears to fulfil your requirements.
Here are two links regarding Linux support:
Amazon.de appears to list it for under 150€. E.g.
Asus Xonar DX 7.1 is on sale at Newegg for $75+ $4 shipping - $10MIR = $69.
PCIe x1 interface? Yes.
at least 5.1 channels? 7.1
24-bit audio output? Yes
at least 96KHz sample rate? 192KHz
high SNR (at least 108dB)? claims 116dB
Response: No, I don't have first hand experience with this card, but I do with another Asus (external) sound product, Xonar U7, ...
When given the choice between onboard, Xonar U5¹ and Prodigy Cube² under $100 one would probably go for the highest quality product available. A few years ago I would have chosen the Prodigy Cube, because it has designer components and an impressive professional looking product page. I bit later I would have gone with professional recording interfaces, but ...