I am in your same situation. Because I use tubes for stereo and solid state for HT, I prefer running two separate sets of cables, one set from each amp, one for stereo and one for HT. I use bananas and plug in the amp I want to use at the time. I don't think you will find a reverse switcher any where and it is very true you do not want to hook both amps to one set of speakers, bad news. Bananas allows a quick connect and it's really not that inconvenient. I've been doing this for some time now and the connectors are holding up fine. I'm using pretty good cables though, Siltech for stereo and Transparent for HT. The Siltech are holding up much better.
Another way, since you are getting a new stereo integrated any way, is to look for one with a "theater bypass" feature/input. What this does is allow you to only have the stereo amp hooked to the speakers, you connect the main preamp outputs of the Insignia into the "theater bypass" input. When this input is selected it will allow the stereo integrated to be a slave to the receiver. The input signal will bypass all preamp controls. Of course, this is a dead issue if the Insignia don't feature preamp outputs. Your best bet then would be the manual switch of cables AT the speaker.