Firstly, a huge thank you to everyone who replied. I appreciate you taking the time to try to help.
Jontait2, your answer was so incredibly simple yet absolutely perfect! I opened the Colourmap dialogue (rather than Colour Palette), dragged it out a bit, pressed Alt+PrtSc, and my Ubuntu OS automatically saves a screenshot into my pictures folder. I've attached a sample image of what it created, and it's exactly what I'm looking for. This is so simple I can't believe I didn't think of it myself!

I appreciate the other comments about using Scripts, but to be honest, the solution above is all I need. I can share this image and use the eye dropper tool to get exact colour values.
Thanks again everyone! You've been really helpful.