i've read a few moms who put their moses basket in the crib to help their babies transiton, so as long as your basinette can't tip over or anything in the crib i'm sure you're fine.
i know you're frustrated with her crying. there were so many times i would also be crying/ready to scream when my babies wouldn't stop crying. i wouldn't leave mine to cry either.
do you have a wind-down routine yet? if not, that's a great place to start. it can include things like dimming the lights (btrw, if her room is brighter than yours is, it might be bugging her), swaddling, snuggling for a moment, turning on white noise, etc. this is something that you will do every time you are getting her ready for bed. eventually, she will learn these cues mean that it's time to sleep.
pat/shh is a great technique to help them learn to go to sleep & then stay asleep, but you need to find the version that works best for you. i learned with luke that he hated the pat part; he wanted my hand to rest gently on his chest/tummy while i shhhed.
unfortunately, crying is to be expected at first. for 5 weeks, she's gotten used to one way of doing things, and now you're changing them. the most important part is to stay consistent. don't leave her to cry, but don't bring her back into bed with you either if you really want her sleeping in her own crib. this makes for a really tough few days, but if you hold your ground she will start to turn around.
good luck & HTH