I can't be sure, but I have a feeling this has to do with enrolling your Mac correctly, first.
I'd simply clicked on the "View the macOS public beta in the Mac App Store" link on the beta.apple.com page, which took me the App Store on my Mac, and from there I'd simply clicked the Download button - which downloaded the initial part of the installer on my machine. But soon after hitting "Continue" on the installer, it threw up the error you're talking about.
I went back to the beta.apple.com web page and enrolled my Mac properly (by downloading and installing the macOS Public Beta Access Utility properly.
Once done, I tried the download again from the Mac App Store (after deleting the intial Installer file in my Launchpad), and voila, the download seems to have begun happening properly.
Suggest you both try that - hopefully, it'll do the trick for you too!
Best of luck!