Regarding requires free disk space Apple says:
If upgrading from macOS Sierra or later, your Mac needs 26GB of available storage to upgrade. If upgrading from an earlier release, your Mac needs up to 44GB of available storage.
You asked "Can I do that, and after upgrading the OS, bring that 32 GB folder back to the Monterey OS library and have the messages just the way there were in Catalina? "
Maybe, but I doubt it. I looked at my ~user>Library>Messages folder and can't find any files that appear to be Messages. My Message file is only 267MB.
Some users have experienced problem with skipping macOS versions. I would first upgrade to BIg Sur then go to Monterey. To get Big Sur
Use Safari to download macOS on your Mac
Safari uses these links to find the old installers in the App Store. After you download the installer from the App Store, the installer opens automatically.
- macOS Big Sur can upgrade Catalina, Mojave, High Sierra, Sierra, El Capitan, Yosemite, Mavericks
I would only move actual data to an external disk
Have you looked at for how to free disk space?
Free up storage space on your Mac - Apple Support
The Store in iCloud might be the best but you will like have to pay for additional iCloud storage since the fee 5 GB you get is not sufficient.
