Burgas has an international ticket office that should be able to book this:

https://www.bdz.bg/en/a/office-for-international-rail-tickets

There is exactly 1 though coach (with couchettes) from Bucharest to Halkali. It runs until 3 October (from Halkali until 1 October):

https://p.bdz.bg/www/s/o/sofia-istanbul-sofia-kor-16082022-ok-3200.pdf

A normal couchette comprtmt has 6 beds-but perhaps this TCDD/Turkish car onle 4. IF you want it all for yourself, you also have to pay for the places that remain empty-otherwise other travellers can booked in. And even that is no 100% garantee-the railway is allowed to sell the empty space to others if there is an overflow of passengers-you will in the end get reimbursed for that-with luck +patience

There will still be much faster, cheaper and a bit less comfy direct buses between the 2-probably fewer though as the many pre-covid.