I heard maybe it should be possible to buy just the reservation from DB and the ticket in Budapest at the station?
Yes, it will be a smart choice, since the price of the reservation is the same either you buy it via DB terminal or at MAV ticket counter in Hungary. While ticket prices are different, because in Germany (or elsewhere besides Hungary/Russia) special MAV-RZD discounts aren't applied.
As of now 2/3 reservation Budapest-Moscow costs 32 euro, 2nd class ticket Budapest-Moscow (when bought in Hungary) is only 117,8 euro