You could install sidebar anywhere or the other so you can toggle to collapse by default. That or write a conditional for your themes that tell them not to display the sidebar under a certain width.
If I have time later I will see If I can figure something out. There might be something already done for that as I think I remember a post about it a while back, I just can't remember..
Not tested but it should work.
p/s: If there is any element such as html divs or graphic or banner that has longer fixed width defined it may hold the page from responsive.