Adding inverses to a symmetric monoidal category (Reference?)

There are left adjoints to the forgetful functors from compact categories (what you call 2CMon) to traced symmetric monoidal categories, and from traced symmetric monoidal categories to symmetric monoidal categories. Composing them should give what you want. The former is known as the Int-construction, and is due to Joyal, Street, and Verity. Sources include Traced monoidal categories and Abstract Scalars, Loops, and Free Traced and Strongly Compact Closed Categories (by Abramsky).

Maybe you could be interested in the recent PhD thesis:

Une introduction élémentaire au 2-groupe de Grothendieck by C. Drugmand, 2016, UCL, Louvain-la-Neuve.

http://hdl.handle.net/2078.1/176774