OK, the solution was actually simple. The NAS is a Synology, so
* Use AudioStation (instead of Media Server or other DLNA server), and
* DS Audio, Synology's iOS remote for AudioStation
The iPad app will let you select any player it can find on your network. It seems that, unlike the controller apps I tried, the playlist is maintained on AudioStation, so you can background the app and tracks still play.
I'm happy(er)! Volume control doesn't work but I'm guessing that because I'm taking the digital output from the Oppo not the analog.