is there a simple tool to determine if my combo of video card and TV is HDCP compatible?

Cyberlink has a free utility that's intended to assess Blu-Ray and 3D playback readiness. While you're asking about the more general case of HDCP compatibility, period, the requirements are essentially the same and this will even break down component by component what the problem is.

The nVidia control panel will display the HDCP status of your card, cable, and display. You'd still need another tool to check the drive and anything else, though.

Have you recently added any drivers? I think windows disables HDCP when certain drivers are loaded because the drivers could access decrypted data and can't be trusted to not abuse that privilege (unsigned drivers, or if you're running windows with driver signing disabled).

I was looking for something like this to test if I could watch purchased Amazon videos in HD, but I didn't really want to download and install anything to my machine.

Amazon actually has a free video linked to in their documentation which can give you an idea of whether your setup supports HDCP. One thing that it doesn't seem to do is explain why your setup doesn't support it. (I think in my case it's my dual monitor setup again.)

December 2020 update

The text of the now removed documentation page, from archive.org:

HDCP for Amazon Video on Computers

You can only stream Amazon Video content in HD through your computer's web browser if your computer and connected display screen(s) meet HDCP requirements.

You can check whether your computer is HDCP-compliant by playing this free movie. Use the video player's HD indicator to determine if your device meets requirements:

• If the HD indicator is illuminated, HDCP requirements have been met and the video is playing in HD.
• If the HD indicator is dim, HDCP requirements have been met, but the video isn’t playing in HD because your Internet connection isn’t fast enough.
• If the HD indicator is disabled, HDCP requirements have not been met and the video is playing in SD. Hover your mouse over the HD indicator to see a message explaining why HDCP requirements have not been met.

Note: Some Video Graphics Array (VGA) and Digital Visual Interface (DVI) connections and monitors don't support HDCP, which may prevent your computer from meeting HDCP requirements.

For more information, go to System Requirements for Streaming on Your Computer.

So if you can find a free, HD video on Amazon Prime Video, it sounds like you can verify HDCP on your machine.