Hi A.,
I just went through a similar situation - 5 1/2 month old boy w/ acid reflux who also had a lot of trouble sleeping. Still has reflux but it's getting better. He used to take all his naps (and some of his night sleep) in the swing. And I would have to get him in a deep sleep at night in my arms in order to get him into the crib (and he was getting way too heavy for that!)
We decided to just bite the bullet and transition him to the crib all at once. I think 5-6 months is a good age to do this. We started on a Friday night, and for every bedtime/naptime we just put him in the crib. I was SO sure that he wouldn't nap. But he surprised me! He cried the first night for 30 minutes (we let him cry but checked on him every few mins). He fussed at naptime for the first few days. But he slept! And after about 5 days he has slept in the crib without any issues, and he's even sleeping longer than before!
Maybe I got lucky, but I would recommend you just give it a try. I had to listen to some fussing and crying, but only for a few days. Just be consistent and don't give in - she may surprise you.
Oh, and we also rock him til he was pretty sleepy, but still awake, and then put him down in the crib. He learned to soothe himself to sleep when he had his normal middle of the night/nap awakenings, which was also a plus.
Good luck!