The app is wrong. There are no seats to Istanbul and they wouldn't be free. From Bucharest, there's only 1 couchette coach.

Booking a reservation online is not possible, only regular tickets can be booked online. Since there is no pass quota, that means you can see availability of the CFR website. Buying a reservation is only possible in Romania. Currently there are still places available.

If they can't sell a reservation anymore at the ticket office, then ask the couchette coach attendant (you'll probably have to pay cash). If that doesn't work, then you can take the same train on a seat to Dimitrovgrad (you'll have to change in Gorna Orjahovica). In Dimitrovgrad, the carriages from Sofia are attached, so you can again ask the their attendants if there are any free places. Only couchettes and sleepers continue to Halkali, so you'll need to get a place there.

I see you have activated your journey already. If you have a flexipass, that not a good idea. The best is not to activate anything in advance, certainly not if you first need a reservation. If you have a flexipass, just cancel the travel day and only activate it shortly before boarding.

So to clarify, it is possible to get on the train on the day, from Bucharest to, say, Tulovo, without needing a reservation? Or do I need a reservation, but it’s easy to get one on the day from the station?

You do need a reservation. 

But the through coach is just a single couchette carriage that is attached to three different trains (or it is the train) on it's way to Turkey. Going to Tulovo would mean arriving at 19.59 and not going overnight (seating carriages from Bucuresti are heading for Sofia and Varna). Don’t know if they are willing to sell a reservation for this at the station at all; worst case you need to fake a journey to Turkey. (At the moment the connecting train is scheduled to run without seating carrages; but the timetable is just for the next few weeks.)

Of course it may also be sold out; you do never know and it depends on travel date. Summary: It will be interesting.

So would I be changing trains at Gorna Orj.? According the app, the train goes direct through (or is it only the sleeper which goes direct through?). 
 

and when you say ‘fake a journey’ to Turkey, do you mean I should ask to buy a reservation for Bucharest to Halkali with a couchette?

thanks for your help!

The train from Bucuresti consists of a 2nd class carriage to Sofia, a 2nd class carriage to Varna and a couchette carriage to Halkalı. Only the couchette carriage goes directly where you’d like to go. 

Normally the couchette carriage is attached to an other train at Gorna Orjachowiza to Dimitrovgrad (with 2nd class carriages) but this one is not shown by BDŽ so it may be just a single couchette carriage running (but it’s shown just until July 23rd at the moment). But if this is true, even changing trains won’t help.

I don’t know if they’d sell a reservation for a couchette when not going overnight; just if this is the case it may be an idea to do a reservation to Turkey.