I'm in the exact same boat now and have been in the past. I have a LOT of experience with your situation
How can I convince HR or a medical professional that X graphics card and Y monitor is the only comfortable environment for me?
The long and short of it is, you don't have to convince anyone. You need to go to a Doctor and explain that the current work environment is causing you pain. He will write a doctors note explaining that you have a sensitivity to the computer equipment at work and need an accommodation.
You then take that letter and you file an ADA (Americans with Disabilities Act) request for accommodation. Your employer will be required by law to provide you with any equipment (different laptop with a different graphics driver and a new monitor that does not cause you pain) you need to accommodate you. They key here is that it cannot represent an "unreasonable burden" to the employer, which a laptop and monitor are not. If they refuse, you can then initiate action under the ADA, which has several stages to it.
I HIGHLY suggest you go to www.askjan.org, it is a website set up entirely to help people who suffer from disabilities navigate the ADA and how to get accommodations. It is a 100% free, federally funded resource for people facing this exact situation, and you can call toll free and talk to someone there who will explain everything you need to know. I would also STRONGLY suggest you download their sample request letters and use them verbatim when requesting your accommodation.
Yes, it sucks, and yes, you will feel confrontational, and yes you will be scared about retaliation (which is illegal BTW), but you have rights, and your employer cannot ignore those rights out of convenience. Anytime you feel weird about it, just think, this is no different than a person in a wheelchair needing a ramp built. No difference at all. And, in my experience, most folks are very empathetic and glad to help people who suffer. The downside of having an invisible disability is that no one knows it's going on until you tell them
Good luck and I am here if you have any other questions.