  Terrestrial exoplanets such as TRAPPIST-1e will be observed in a new capacity
with JWST/NIRSpec, which is expected to be able to detect CO$_2$, CH$_4$, and
O$_2$ signals, if present, with multiple co-added transit observations. The
CO$_2$-CH$_4$ pair in particular is theorized to be a potential biosignature
when inferred to be in chemical disequilibrium. Here, we simulate TRAPPIST-1e's
atmosphere using the ExoCAM General Circulation Model (GCM), assuming an
optimistic haze-free, tidally locked planet with an aquaplanet surface, with
varying atmospheric compositions from $10^{-4}$ bar to 1 bar of partial CO$_2$
pressure with 1 bar of background N$_2$. We investigate cases both with and
without a modern Earth-like CH$_4$ mixing ratio to examine the effect of CO$_2$
and CH$_4$ on the transmission spectrum and climate state of the planet. We
demonstrate that in the optimistic haze-free cloudy case, H$_2$O, CO$_2$, and
CH$_4$ could all be detectable in less than 50 transits within an atmosphere of
1 bar N$_2$ and 10 mbar CO$_2$ during JWST's lifespan with NIRSpec as long as
the noise floor is $\lesssim$ 10 ppm. We find that in these optimistic cases,
JWST may be able to detect potential biosignature pairs such as CO$_2$-CH$_4$
in TRAPPIST-1e's atmosphere across a variety of atmospheric CO$_2$ content, and
that temporal climate variability does not significantly affect spectral
feature variability for NIRSpec PRISM.
