In Adjoints to functors from categories of algebras Rattray defines that a category $\mathcal{A}$ has LAP (left adjoint property) when every continuous functor on $\mathcal{A}$ has a left adjoint, or equivalently every continuous functor $\mathcal{A} \to \mathbf{Set}$ is representable. This is exactly the dual notion to what I called "strongly compact". So Rattray would probably call this property RAP (right adjoint property). He shows that every monadic category over a RAP category is again a RAP category. (He proves the dual version: every comonadic category over a LAP category is a LAP category.)

It is a bit confusing that it is claimed in the mentioned paper "Compact and hypercomplete categories" by R. Börger, W. Tholen, M. B. Wischnewsky, H. Wolff that Rattray actually proves that every monadic category over a compact category is again compact: compactness involves hypercontinuous functors. But probably Rattray's proof can be used in both settings.