$E_6$ inspired SUSY models are a well-motivated class of models
that can solve the $\mu$ problem in the MSSM and allow for the
tree-level Higgs mass to be increased. In the simplest variants
of these models, multiple exact and approximate discrete
symmetries must be introduced for the model to be
phenomenologically viable. Here we study a constrained version
of a recently proposed $E_6$ inspired model, the CSE$_6$SSM, in
which only a single, exact custodial symmetry is required. The breakdown of
$E_6$ also leads to an automatically conserved matter parity, implying the
existence of two dark matter candidates in this model. We demonstrate that,
while one of these can be extremely light and contributes negligibly to the
relic density, a predominantly Higgsino or mixed bino-Higgsino MSSM-like
neutralino can account for the observed relic abundance of dark matter.
However, recent limits from direct detection searches place severe
constraints on the mixed bino-Higgsino scenarios that account for the full
relic abundance. We show that a large part of the parameter space with a
predominantly Higgsino dark matter candidate will be discoverable at
XENON1T. In these scenarios, the exotic states can still be light enough to
be observable at the LHC, allowing for the discovery of the model in
collider searches even in scenarios with an otherwise heavy supersymmetric
particle spectrum.