Divergence of finitely presented subgroups of CAT(0) groups

We construct families of $CAT(0)$ groups containing finitely presented groups whose divergence functions are of the form $r^\alpha$ for a dense set of exponents $\alpha \in [2,\infty)$ and $r^q\log(r)$ for integers $q \geq 2$. The same construction also yields examples of $CAT(0)$ groups containing contracting elements which are not contracting in certain finitely presented subgroups.