Hi, we went through this would only keep awake for an hour at a time, I got so worried I got the health visitor out. She is 7 months now and stays awake alot is down to 2 naps and a catnap if she needs it. So it does get better.
Only wanting to sleep in cot, we too have this however it is getting slightly better. Does your dc sleep in a swaddle I have found that if I make sure she is cozy and felt like she is wrapped up she will sleep. When we have cold spells I make sure she is wrapped in a snowsuit, another thing I have is called a 'bug in a rug' and she always falls asleep when she is in that. I think it is something to do with the fact that she sleeps in a grobag/sleeping bag in the cot. Also my dd is a tummy sleeper so when she is in the car seat or pushchair she can't do that.
Just wanted to let you know you are not alone!
It will get better, then unfortunately it will be something else! Sorry