Is there any way to make multiple http requests with a shared session using the

No. But you can save the needed information to PVs in the first step and reuse that in the second. I do that with webservices to access a system that has a special login function that returns a token that has to be used in all following calls.

If you can run the node on MNI (within a subprocess) and specify run one node at a time where you build your logic accordingly, it might work. I haven't personally tried it but seems like a possibility.

Unfortunately that's a slightly different use case than what I have. Though I am also authenticating with a service it relies on a session token stored server side - there's no token in the return body or headers.
Sathya: An interesting idea however I checked and each invocation of the node creates a new session, no luck there.