  We give a quantification of residual finiteness for the fundamental groups of
hyperbolic manifolds that admit a totally geodesic immersion to a compact,
right-angled Coxeter orbifold of dimension 3 or 4. Specifically, we give
explicit upper bounds on residual finiteness that are linear in terms of
geodesic length. We then extend the linear upper bounds to hyperbolic manifolds
with a finite cover that admits such an immersion. Since the quantifications
are given in terms of geodesic length, we define the geodesic residual
finiteness growth and show that this growth is equivalent to the usual residual
finiteness growth defined in terms of word length. This equivalence implies
that our results recover the quantification of residual finiteness from
\cite{BHP} for hyperbolic manifolds that virtually immerse into a compact
reflection orbifold.
