3 month old won’t sleep more than 3 hours at night if I’m lucky

My baby is breast fed. She is 16lb (or more) and 3.5 months old... she has never been a good sleeper. Sometimes I get 3-4 hour stretches if I’m lucky the past month or so . She did sleep 6 hours straight twice so I know she’s capable of it and those were both times she had formula before bed when my mom babysit her at 2 months old. We thought maybe formula would help so tried powder and liquid but neither did anything ! She woke every 1-2 hours. So now it’s back to breast feeding.

I’ve been doing a routine with her since she was 4 weeks old and that includes bath, lotion, feed, pjs and lights off etc.

Lately she will fall asleep then wake 30 mins after aand screams. Crying uncontrollably. She will only settle if she’s breast fed and laid with. It’s impossible to get her to sleep in her crib and I’m up 5+ times a night with her so I don’t get any sleep unless she’s in bed with me. My boyfriend and I now sleep in different rooms because he works at 3 am and needs sleep.

I’m so sleep deprived and it’s affecting our relationship. She passes gas just fine and poops everyday or every second day at least. She’s an awesome baby otherwise but still tired a lot in the day and tends to get over tired as she’s hard to get down for a nap but most times I can catch her drowsy and she will nap but never really more than 30-60 mins at a time.

she’s also been put in her crib awake many times and falls asleep on her own yet at bedtime she will not and she can’t go back to sleep when she wakes. I feel like the cry it out method isn’t even possible for this kid. She cries until she chokes and shakes. I’ve let her cry for an hour (checking in every 5-10 mins) and it worsens as time passes .

Any ideas ?

Also paid $50 for a zen sleep sack that did nothing