I think the only way to run web.config transformations locally is to use MSBuild with the following parameters, where [YourTransformationConfiguration] is the name of the configuration to use, e.g. Debug or Release:
msbuild PathToProject /T:Package /P:Configuration=[YourTransformationConfiguration] /P:AutoParameterizationWebConfigConnectionStrings=False
The output of which will be available at:
[ProjectFolder]\obj\[YourTransformationConfiguration]\Package\PackageTmp
From here you can run migrate.exe on your assembly, which is similar to running the Update-Database PowerShell script. If you used Nuget, this will be in the tools folder of the EntityFramework package. I had some trouble getting this up and running, but there's a useful answer Stack Overflow answer here:
Getting Migrate.exe to work