On the face of it, it certainly sounds like it may be a REIN issue.
I presume you have gone through all the usual troubleshooting such as testing from behind the master socket etc?
Since you have had no luck tracing the cause of REIN yourself, the only thing I can suggest is that you monitor and present the graphs to your ISP as proof of suspected REIN... and hopefully they can then get a specialist REIN engineer out to you. Unfortunately REIN can be very hard to track down, and you may have to jump through a few hoops first before someone at your ISP/BT takes note.
Good luck.