I have not done this before but for external integrations usually a web service is used. (REST, OData etc.)
However if you intend to call a native API then going the Java way is the best route according to me.
So, did you explore call your .NET assemblies from Java directly already?
because if i search online i keep encountering blogs, threads which speak about JVM and CLR clashes.
I interesting stackoverflow thread explains it well. Take a look for some inspiration.