Would it be possible to create a "placeholder", something like a certificate or special paper for the quest completion? Meaning, you accept the quest, it gives you your tasks AND a certificate (looks like a quest paper or whatever). If the task is "kill 20 ____" then after you finish killing the 20 whatevers, you need to put the certificate in your inventory, and THAT (placing in inventory) is what finishes the quest. Then, the quests "takes" the certificate away, and replaces the spot with the reward bag.
My fear is that if you just give a bag of items as the reward, the same thing would happen if your inventory is full as it would normally.