I am not an expert on NuGet and have only limited experience with "nuget.exe", but I am pretty sure I understood what your actual problem is, and I think the following may be a conceptual solution:
On your build server, MySolution exists in two versions in parallel:
the last commited version
the "NuGet" version (which corresponds to the last time a feature ...
If you are going to test this way you are going to need to be more nuanced about it.
I'm going to presume that the actual library being released to nuget must pass these tests that you are releasing as some form of acceptance strategy.
If that is the case:
the other team builds, and tests their nuget package.
given they are happy with it they release it ...