I tried to upgrade my late-2012 iMac from Yosemite to High Sierra, but when I went to the app store I received this message:
This copy of the Install macOS High Sierra application is damaged, and can‘t be used to install macOS.
Is there a solution to this, or do I need to put a new iMac on top of my Christmas list?
I'm a bit confused, are you saying the App Store told you the Install macOS High Sierra app is damaged? Or do you mean after you downloaded it you got that message on launching the app?
krs, is correct, that message is often given if the certificate for the app has expired but that should not happen if you just downloaded it from the App Store.
It seems clear from what I have read that the late 2012 iMac does have problems running Mojave so High Sierra is as far as I would recommend you go.
Here is a very handy link from Apple with instructions and links to upgrade the macOS on older Mac
https://support.apple.com/en-us/HT211683
Note: If you are using the links that require using a web browser it is best to use Safari as krs has already suggested.
If you continue to have a problem with the El Capitan installer you could try installing Yosemite then upgrade from that to High Sierra.
Let us know how that works for you.