...but the 'real' compositions on vagonweb.cz are showing that it has run without a sleeper or a couchette.
Well, "real" composition dates Thu 6.2. / Fri 7.2.2014, it was quite long time ago. Maybe then there was some reason why carriages were missing, but as of now MAV website easily allows to purchase ticket+reservation sleeper/couchette
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So in this case we would recommend to trust official MAV website, hardly they would sell reservation so easily while knowing that there is no carriage(s) running
Is it possible to buy a 1st class special ticket for this train?
Since there is no "Belgrade Spezial" for 1st class, it's not sold online. But at the station it won't be a problem to buy such ticket (fare should be 42,7 € one-way or 48,8 € return)