I think I found the answer. Constrained delegation is the feature of the Kerberos protocol I expected to exist.
If we use SSL/TLS with mutual certificate based authentication, then the client will be authenticated by the proxy, who validates client's certificate by a local CA (within the hidden intranet). Afterwards, the proxy will generate Kerberos tickets on behalf of the already authenticated client.
At the server side, the ticket validation should happen at the runtime level (e.g., by IIS).
Hence, if the client is able to consume the service through SSL/TLS, then the Kerberos authentication remains fully transparent for client and the server.