He then fell to 27th in Race 3, which saw Williams rise from 13th to ninth in a race won by Frederik Vesti.
Three different winners claimed honours across the weekend, including David Schumacher sealing his maiden success in the category in Race 2.
That came a race after the German was involved in a clash with Williams at Turn 4 in Race 1, for which the Aussie was penalised five seconds, dropping him back to 19th at the flag.
However, a swathe of penalties saw him rise to 16th in the final classification.
Race 1 winner, Dennis Hauger, rose nine places in Race 2 to claim another podium position, while Doohan also climbed three spots to finish seventh.
However, his weekend then took a turn in Race 3 after he tagged the back of Jak Crawford's car while battling for fifth.
That dropped the pair down the order and triggered the Virtual Safety Car, Vesti going on to win with Hauger again on the podium.
The Norwegian has now increased his championship advantage over Vesti to 31 points, with Doohan third, two points further back.
Round 4 of the FIA Formula 3 Championship is set to take place at the Hungaroring at the end of July.