I guess your Julia package mainly does numerical computing, right? In that case, I believe that it is eaiser “to use a messaging protocol” through some communication mechanism like inter-process communication (IPC). Preferably, you may try the ZeroMQ library. You can find more discussion in this thread Running Julia from JAVA? What is crazier?. Just replace Java with C#.
In short, your Julia code may set up a server and listen for requests from the C# client, e.g., the request “svd” (along with data) may require the Julia code to perform some SVD on the given data. Then, the Julia server sends the solution back to C# client. MsgPack can be used to encode the message communicated by ZeroMQ.