Thanks for the tip. However, I managed to avoid attaching TNObjects to missiles: I simply sync every player's mouse position along with their other data, then everyone spawns a local copy of the missile which follows the - supposedly - same logic and follows the same position. Of course, only the owner determines whether or not a successful damage-delivering hit exists, so even if some clients see it a bit differently, I'm sure I can live with it - it's a missile with a blast radius, after all, and not a precise shot which can be easily avoided.
It works, seemingly well-enough. Now it's time to fix up a team selection screen of some sort.

On a completely unrelated note, I was going to ask about an "issue" I just spotted. Given that all my player entities use physics-based controllers, I'm making good use of TNSyncRigidbody amongst other things, and I update once every two seconds. However, I just noticed that I seem to be getting noticeable "jitter" once every two seconds (my TNSyncRB update rate) on instances that are NOT the Unity Editor when my Transform gets repositioned. This had never happened before.
After a bit of confused fiddling, though, I noticed that I'd switched it from UDP to TCP (as I did want my transform repositions to be sent reliably); switching it back to UDP fixed the issue. I'm not quite sure it was an issue in the first place, but it's gone now, so I'm content.