As per my knowledge, It might require some custom scripts for this fucntionality to work.
I suppose one way to do it using OOTB rules would be to use a wrapper harness around the harness you want to launch and include a section containing only the menu on top of that harness. Call this wrapper harness in the nav rule.
In that case when the harness is replaced on click of the menu, the whole harness will be replaced with the new harness which also contains the menu. This is not the exact functionality as you mentioned, but it should serve the purpose.