Hi,
Lovely to meet you, however, I hate to be the bearer of bad news.
The HDMI to USB-C adapter is never going to work. They're meant to work the other way, and you can't split HDMI for multiple distinct displays. You can get add-in cards to give your computer a port to plug the dock into, but I've looked at the spec sheet for your desktop, and I'm pretty sure it doesn't support any of those cards. I think your best option is to buy three monitor switches online - they are about £25 each on amazon - and stick them under your desk. You can press each button to swap the display source when you want to switch. It's a bit clunky, and you'll have to hide nine display cables under your desk, but they could also switch your USB devices, like your keyboard and mouse, if you use them like that.
You could check if your monitors support chaining. Some DisplayPort monitors can be hooked up to one another, so you only need one cable going to your PC. Then you could just have the one switcher.