  The exploration of the redshift drift, a direct measurement of cosmological
expansion, is expected to take several decades of observation with stable,
sensitive instruments. We introduced a new method to probe cosmology which
bypasses the long-period observation by observing the redshift difference, an
accumulation of the redshift drift, in multiple-image gravitational lens
systems. With this, the photons observed in each image will have traversed
through different paths between the source and the observer, and so the lensed
images will show different redshifts when observed at the same instance. Here,
we consider the impact of the underlying cosmology on the observed redshift
difference in gravitational lens systems, generating synthetic data for
realistic lens models and exploring the accuracy of determined cosmological
parameters. We show that, whilst the redshift difference is sensitive to the
densities of matter and dark energy within a universe, it is independent of the
Hubble constant. Finally, we determine the observational considerations for
using the redshift difference as a cosmological probe, finding that one
thousand lensed sources are enough to make robust determinations of the
underlying cosmological parameters. Upcoming cluster lens surveys, such as the
Euclid, are expected to detect a sufficient number of such systems.
