  We give a definition of an amenable fusion module over a fusion algebra. A
notion of relative integrability for the `coduals' of coideals of compact
quantum groups was recently introduced in the joint work of de Commer and
Dzokou Talla. We use this property to construct an analogue of the
quasi-regular representation. Then, we characterize a certain coamenability
property of quasi-regular representations with amenability of their associated
fusion modules. Afterwards, we obtain a duality result that generalizes
Tomatsu's theorem for this coamenability property and an amenability property
of their `codual' coideals (under an additional assumption). As an example, we
apply this result to show the fusion modules associated to certain non-standard
Podle\'s spheres are amenable.
