I have a somewhat extreme suggestion. Perhaps you could wrap the third party code with something that exposes it as a SOAP or other web service, then use standard libraries (SOAP::Lite, etc.) to hit it?
Actually that is not all that extreme a suggestion, two approaches are available: either to explicitly create an ASP.NET web services facade for the .NET APIs or to create a small wrapper that exposed them using HTTP Remoting. The former is probably the more flexible and depends less on the way that the API has been designed. The great advantage of using SOAP rather than, say, COM interop is that you are not limited to having the Perl parts of your application run on Windows.