The model of exponential fields seems to be related, but these are defined with an exponential function instead of a commutative operator, and no inverse function.
Also, many of the ideas outlined above where first proposed by @goblin back in 2013, but only in the context of $\mathbb{R}$. One of the main benefits of the Hypertype Theory outlined above is to provide a recursive sequence of larger and larger objects that are isomorphic to subsets of $\mathbb{R}$.
Furthermore, if my understanding of Schanuel’s conjecture is correct and if the conjecture is true, then $\pi$ is not a term of $\mathbb{H}_n$.