Same here, I have perpetual licenses of Project 2013 and Visio 2013, and am also an Office 365 subscriber.
Spent about an hour chatting with 2 MS reps. The first one confirmed that there was currently no way to run Office 2016 with Visio/Project 2013. After pointing out the problems, he did say that there was an option for MS to (presumably) remote in and manually
install Project/Visio 2016. He transferred me to a second department.
Second department (after going through the explanation) said that this wasn't possible. I was assigned a case number, and told that a solution was forthcoming (no ETA, however).
I would truly appreciate someone from MS reaching out to help get this situation resolved (or, better yet, offering a concrete, authoritative statement on what MS is going to do to address the situation and what the timeframe is).
-GW