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 necessary and so forth. This is particularly true for games where the soundscape is rendered by the game engine and it doesn't get more accurate with 2 speakers or headphones than the game engine knowing that it should render a stereo signal. A multichannel downmix by an unknown element in the chain doesn't improve the result it's just a dishonest, technophobe marketing myth and number game that is still around.
Stereo (1), generated Downmix (2) and 5.1 Mix (3-8) of Paschendale from Iron Maiden (Sources: 24-Bit FLAC, DVD-A)
If you need a visual example, download Audacity or any other audio editor, load a multitrack audio track from a movie or audio recording and mix it down to stereo. Many people pretend to care much about clean audio. Does the downmix with information from all channels look cleaner or more detailed to you? No, it doesn't! Good games are able to mix all the information into a high quality stereo signal. With the rise of VR binaural audio will get more important and virtual 7.1 mixes will hamper your immersion. Please let game engines do their work without interference, the game designers who put them together should know better how to treat audio that they or their customers create.
I could suggest to you to get a cheap 7.1 USB soundcard from China, but that's worse than your built-in sound solution. A multichannel audio production interface can actually deliver good quality but is at least three times out of your budget, not very portable and the Windows drivers don't support running the channels as a 7.1 configuration for gaming while the hardware is capable of doing so. I'd rather recommend an O2+ODAC or a similar product and reference headphones if you're already willing to spend +150 USD.
Here is a review from a Razer fan who isn't very impressed with this headset either. I found this video while looking for a picture of the connectors.