If I understood correctly, you are asking whether the spectral gap of a non-reversible Markov chain gives any general control on its Poincaré constant (which is the spectral gap of the additive reversibilization). The answer is no, even on finite state spaces: consider the Markov chain on $\{0,1\}^n$ which, at each step, replaces the current state $x=(x_1,\ldots,x_n)$ with either $(x_2,\ldots,x_{n-1},0)$ or $(x_2,\ldots,x_{n-1},1)$, each with probability a half. This chain has spectral radius $0$, i.e. spectral gap $1$. Yet, its Poincaré constant tends to $0$ as $n\to\infty$ (like $1/n^2$).