I am currently trying to find a work around to build and publish locally a .sqlproj project. Apparently this is meant to be done using SSDT tools provided by Visual Studio, which is not available on Linux.
My question is, what are my options? What can I do to build and publish this project type?
You don't need to have Visual Studio if the only your intention is to publish the project. You need just MsBuild and probably SSDT components. This is discussion about MsBuild on Linux.
Another way is to use docker, for example this one.