Sounds like it's all on one itinerary/ticket, which they wouldn't sell if it was invalid.

This being the case, if you missed the connection, they would put you on the next flight. There appear to be 3 flights a day BUD-IST and IST-TBS, so as long as the arrival time of the next flight is also acceptable I'd go ahead and book it.

I'll be experiencing BUD (airport) for the first time tomorrow so can report on the practical aspects of transiting there after, but maybe someone else will chime in first.