I found that the gradual approach worked for me when dd was little, so I started off trying to get dd used to settling with my assistance but without taking her out her cot...bit of a labour of love though... I would give her a dummy ( she's a very sucky baby) put her on her side (her favoured sleeping position) and pat her bottom and shush her until she fell asleep. It took some persistence though! Then after a few days or so I withdrew the shushing and then the bottom patting and then eventually ignored her and she settled after some short grumbling. To work though I had to go to her before it escalated, so as soon as I heard her starting to moan but before the full on crying. I have no problem with the CC in theory at all but I just couldn't do it when she was so little. She's almost 5 months now and has started being a bit more difficult again and so I give her 10 to 15 mins of yelling. If she doesn't settle after that then I go to her and do the above, but often she's sound asleep by the time I get back from making myself a cup of tea pouring myself a drink . I don't feel too bad about it because when I go to her when she's crying she stops the minute she hears the door opens and gives me her best "yey I've won" smile! So I'm pretty sure it's not as traumatic as she makes it sound.
If you don't fancy that option have you tried playing certain music at bedtime or when you rock her and so try using music to soothe her rather than picking her up? It doesn't work for my little one but I know others who swear by it.
Also, can you try to keep him awake for longer periods so that he will nap better? My dd now has 30 mins at 9am, two hrs between 12 and 2pm (ish) and a short 20min nap around 4pm. I got to this outcome by trial and error - if she has longer than 30mins in the morning she wakes up 45 mins into her lunch nap and is difficult to resettle. If I don't give her a 20 min nap at 4pmish she is a nightmare at bedtime and won't feed properly. babies are all different but from the reading I did they do really benefit from a two HR nap at some point during the day.
Sorry this is getting long - I'm not being very eloquent today! Re the feeding, babies love naturally to snack and in my experience the only way to get some to take a decent feed is to grit your teeth and not feed them on demand but every three to four hrs. Nightmare the first time because he will have only snacked and will be hungry but if you can hold out and distract him with dancing, tv, anything, then the next time he feeds he will be so hungry he'll feed for longer and then in turn will not need feeding again for longer. My SIL had this exatct problem, minus the reflux) and ended up paying a fortune for a very experienced sleep trainer who managed to sort out her dd in a week by using this approach, although I warn you my poor SIL had to leave the house. Her dd is now a smiling happy baby on a good feeding and sleeping routine. caveat all this though because I have no experience with reflux so I don't know what the implications are for feeding so prob best to check with your gp.
Rambling, but I feel your pain and wanted to try and help. Good luck! X