"looking forward to being". Is this correct?
Looking forward is always followed by the +ing (gerund). A simple way to check this is to ask it in a question: "What are you looking forward to?" ...being part of the team.
I think if this were rephrased as, "I'm looking forward to being part of the team so much that I can barely sleep", might sound a bit more natural.
The phrasal verb to look forward to takes a gerund as its object, so the second version is correct.