I pick my lo up and hold her for the dream feed. If lo is in a deep sleep and you pick them up really slowly and gently, they shouldn't wake all the way up, maybe just wiggle a bit. Once I sit down with lo she's prefectly still again and sound asleep and I just touch the bottle nipple to her lips and after a few seconds she'll take the nipple and drink. She keeps her eyes closed and eats while she's pretty much asleep. She's sound asleep by the time she finishes and I don't burp, they drink so slowly with a dream feed that they don't need a burp.
The whole thing seems almost impossible until you do it lol. Now, some babies don't do well with it and it disrupts them, most will be fine though and you have to try it to know. They gets really used to it after a few nights too, part of the routine.
I wouldn't do it while lo is laying down in her cot, you don't want her feeding laying flat like that. Some people do do it that way, but I wouldn't.