I was lead to believe that babies should and do always sleep in their cot...
Loads of babies never sleep in the cot. Ever. Loads cosleep at night, sleep in arms during the day. Some people establish the pushchair for naps rather than the cot.
When I had my first child, I read so much and stressed so much about doing everything "right", that she ended up with really appalling, long term poor sleep habits. I feel certain if I'd have just relaxed, stop trying to fit my square-peg baby into a round hole, that everything would have been so much easier and more straight forward.
My DC1 and DC4 (my two girls) had very similar sleep profiles as babies. They need my consistant reassurance and lots of calm, quiet comfort to sleep. The outcomes of these similar temperaments are polar opposite.
DC2 and DC3 (my two boys) were of the give dummy, put down, leave sleep profiles. I tried to turn DC1 into this kind of baby (But without the dummy, I was too scared to use a dummy with my first). She wasn't that kind of baby and never will be. So I ended up with years and years of poor sleep habits. Compare to DC4, similarly needy in terms attention needed to sleep as a baby. I didn't fight it. Ever. I just followed her needs and gave her everything she needed. It took her longer to get to the "easy" way the boys were. But she got there. DC1 never did
Maybe Dottylottiesmum - your baby is the square peg trying to fit in the round hole. You'll get the best outcomes accepting the type of child you have, rather than trying to turn her into something else.