If you are concerned about burping, my suggestion would be just to hold your little one in the upright position and rub her back. Something to keep her asleep, but also to work out the gas.
The 12:30 could be habitual waking, especially since you did the df last night and she woke up at the same time. I would keep handling it the way you have been doing it.
Perhaps offer more at the df if you feel she can take more.
Also, she may not be ready to go all night without one other feed plus the df. Are you sure she's ready to go through?
HTH