I think it's very early days to be considering this - lots of newborns are unsettled at night and it's a bit early for colic. Could it be that your tiredness means you rush and don't burp as well at night? Or maybe baby doesn't like the crib and needs time to get used to it. I would give it a few more weeks though, and see if things settle naturally.
Our LO used to get quite unsettled and windy and night although he is bf. I was producing lots of milk so he'd gulp like yours. We found he responded really well to bicycle legs and tummy massage, he'd calm instantly and start whinging the minute we stopped. We also started using Infacol. I'd lie him next to me on his tummy at times too when it was bad and he loved that but I wouldn't sleep just watching him!
Put her in it during the day? We bought the Moses basket down stairs in the early days so she could have her naps in it, then she had her naps upstairs in her basket. She hated the basket for the first few weeks but slowly got used to it. Also we put a hot water bottle in it to warm it for her and I slept with a muslin and put that in with her
It's quite early for colic/comfort milk. you need to use a faster flow teat as the milk is thicker and I don't know if a baby that young could cope with it. It is normal for newborns to be more active at night for a while, they are used to being in the dark in the womb. If it was wind only I'd say to try Infacol, but it seems to be more due to the fact she is unsettled/gulping her feed down. Could you try the pram at night to see if it makes a difference?