  We use the general framework of summation by parts operators to construct
conservative, entropy-stable and well-balanced semidiscretizations of two
different nonlinear systems of dispersive shallow water equations with varying
bathymetry: (i) a variant of the coupled Benjamin-Bona-Mahony (BBM) equations
and (ii) a recently proposed model by Sv\"ard and Kalisch (2023) with enhanced
dispersive behavior. Both models share the property of being conservative in
terms of a nonlinear invariant, often interpreted as entropy function. This
property is preserved exactly in our novel semidiscretizations. To obtain
fully-discrete entropy-stable schemes, we employ the relaxation method. We
present improved numerical properties of our schemes in some test cases.
