I am trying to find an example of the following situation.

$G$ is a t.d.l.c. (totally disconnected locally compact) $\sigma$-compact topological group in which every compact open subgroup is topologically finitely generated. $H$ is a maximal compact open subgroup and $K$ is a subgroup abstractly isomorphic to $H$ such that the intersection $H \cap K$ has countably infinite index in both $H$ and $K$.

I believe that I can show that this situation is not possible if $G$ is the group of rational points of an absolutely quasi-simple algebraic group over a non-archimedean local field $k$.

Would welcome any useful suggestions for determining if this situation is possible.