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OK, so I think I solved my own problem:
it appears that Template is ignored when deploying a Design to a pre-existing site. I used the value "1" for each of my Designs and pushed those design to some classic Project Sites and didn't see any corruption. So, either I guessed right or the parameter is ignored for pre-existing sites.
I also found that "Invoke-SPOSiteDesign" did not work for me. I got an error message "Invoke-SPOSiteDesign : The site does not allow site designs.". I initially thought this was due to allowing/preventing custom scripts, but that did not appear to be the case. Even with DenyAddAndCustomizePages disabled, I was able to push the Design to the site using "Add-SPOSiteDesignTask". It takes a little longer and I can't seem to track the status of the job as it's deploying, but it seemed to work.
Right now I'm manually running the command to deploy the set of lists to individual sites, but it's still significantly faster than manually deploying 4 lists to each of the sites.