That sounds somewhat a good idea. Sorry to hear that nothing worked after all the numerous ideas and suggestions myself and others thought of. It's really bizarre that nothing worked but it does sound like the issue is from the Amp's end.
For now, just connect your Optical cable from your PC to your Amplifier and the HDMI from your PC to your TV. Make sure you adjust the sound devices default to the Optical output and your PC see all the audio features that an Optical cable offers/restricts you. Unfortunately, you will not be able to get very high bandwidth sound or even Dolby TrueHD nor DTS MA sound, however, as you do not watch any movies via your PC anyway and most - if not all - games do not feature HD quality level sound let alone uncompressed sound, you will not be missing out anything anyway. Plus, it is very difficult to hear the difference between 320kps 24-bits MP3 and Uncompressed WAV/PCM/DTS HD via a really good amp let alone playing a game that may utilise such high-quality sound samples (which they don't).
So yes, the above is indeed the best course of action for now so you won't need a new Sound Card. However, you will need to purchase a sound card for sure if your machine doesn't have an Optical Output. My only hope is that there won't be any sound lag caused by your amp but I seriously doubt that would happen, especially in consideration that your amp is recent.