Can my site use two different WordPress installs for different pages?

That would be an extremely tricky approach to the problem I think. If your end goal is the replacement of one template with another (a migration), then a more standard approach would be to install an new WordPress instance on a different URL. Perhaps using http://new.example.com (a subdomain).

This is then your “development”, or “staging” server (if a remote server is actually required at all of course.)

Once you replacement work is complete and tested, you would replace the live files with the development files, and complete your migration.