  We consider here asymptotic models that describe the propagation of
one-dimensional internal waves at the interface between two layers of
immiscible fluids of different densities, under the rigid lid assumption and
with uneven bottoms. The aim of this paper is to show that the full
justification result of the model obtained by Duch\^ene, Israwi and Talhouk
[{\em SIAM J. Math. Anal.}, 47(1), 240--290], in the sense that it is
consistent, well-posed, and that its solutions remain close to exact solutions
of the full Euler system with corresponding initial data, can be improved in
two directions. The first direction is taking into account medium amplitude
topography variations and the second direction is allowing strong nonlinearity
using a new pseudo-symmetrizer, thus canceling out the smallness assumptions of
the Camassa-Holm regime for the well-posedness and stability results.
