tl;dr: 5-month old's reflux has improved with my diet changes but her sleep hasn't improved. She MAY still have some reflux symptoms but it's truly hard to tell if it's that or constant cold symptoms. Sleep training appropriate or no? What kind?

We're at our wits end with nighttime sleep and our 5-month old! She's a reflux baby, and so over time we've had to institute lots of sleep associations in order to get her to sleep.

Now her reflux has improved a lot with my elimination of dairy, soy, eggs, coconut, tomatoes, and some other things. During the day she is super chill, happy, and takes great naps. But she doesn't sleep great at night. About 5 out of 7 nights of the week she still has some coughs and congestion that make me think her reflux is still bothering her a bit, OR that she consistently has symptoms of a cold, but I don't think she's in pain. It takes 1-2 hours to get her down at bedtime, and she wakes up 2-4 times per night. She sleeps in the pack n play bassinet, but when her congestion seems really bad she won't sleep in it and we have to put her in the Rock n Play.

Is it appropriate to do some kind of sleep training at this point? My reluctance is that I know it won't work if her reflux is still bothering her, but I honestly can't tell if her sleep difficulties are still due to reflux or if she's just used to waking up a lot. Thanks for reading this far!