At https://github.com/jupyterhub/jupyterhub-idle-culler/issues/8#issuecomment-652481646 there is one possible solution how the cull_idle_servers script can be altered (as well as the JupyterHub config) in a way that could also be applied to this use case.
This and in addition the points of @mriedem sound like a separate service with slightly altered code but a very different name (to avoid confusion for new administrators) sound like an easy and implementable option to me. The only issue with the above-mentioned solution is that it requires monkey-patching and that is not the cleanest of all options.