If you have the resources to do so, try to perform an A interface trace to understand who is performing the LU (which IMSI’s ?).
I experienced a situation where the subscribers froma different operators (but same country) were trying to attach during the night, because their own PLMN was down for a while.
Your competitor is perhaps performing some network upgrade, which cuts the BTS’s in some areas during the night.
SDCCH MHT of 60 to 120 seconds is a little strange though. Here again, an A interface trace could help you understand the matter. Is there anyone in your MSC that could provide you with some info ?