Sechel, BTW, it’s IMPOSSIBLE that the minhag not to sleep in the sukkah dates back to the Alter Rebbe, otherwise he would have pointed it out in his SA.
The whole minhag dating back thing seems to be the Rebbe’s innovation.
As far as not sleeping in the sukkah, I have no problem with that at all, neither does Belz. It’s the Justification I have a problem with. (Many people have minhagim that seem to contradict halacha, but if it’s your minhag you can follow it – unless you went thorough the sugya yourself and feel otherwise.)