I hope you don't mind me putting my ten cents worth in. I really feel for you. It is so difficult when your LO wakes up so often in the night.
My LO is 11 and a bit months old and we left doing EASY until about 10 months or so. It was SO hard at first and it has taken since the beginning of September, but she has finally been able to sleep through the night and has done so for the last three nights. We have had many NWs over the time and sometimes it would take two hours to resettle her, and we have had our day starting at 4.30 am and she doesn't nap much at all. I have been exhausted too, so I really understand that. I have been going to bed at 8.45 pm and to sleep at 9.30 pm every night for a while. I also try and sleep when she does. If you can, perhaps you could try that and catch up on a little sleep.
We were co-sleeping with her until 7th September when we put her in her own room and cot. It was a big move, but she did brilliantly. She used to be fed to sleep as well, but she has managed to stop doing that too. I also fed her in the night, and I stopped doing that as well. I felt that was encouraging her to wake up. I am not sure though at the age of your LO whether you can do that. I think you are doing great with only feeding him every four hours in the night. It is obviously not for that reason he is waking up.
What I am trying to say here is that it will take time, patience and perseverance, but you will get there. I had so much brilliant help from some of the members here and I am sure you will too. Just keep going even when you feel like giving up with it. It is worth it.
I think the NWs may be to do with what is happening during the day. With our NWs and EWs I think it was because she was going to bed too early. I was putting her to bed early because she wasn't napping. I thought that would help but it just created many other problems. I found that stretching out her A times really helped and also setting her naps. I think that is regulating her body clock. We only started the set naps a few days ago, but I am glad we have started it. I felt that I needed some regularity and rhythm. I am finding it great. Over the last few days, we have been tweaking her A times to best suit her and get the best naps. I know that set naps are not strictly BW, but they can work. If you go with whatever A time is appropriate for your LOs age, you could start from there.
When I started this, I held her whenever she cried, then I gradually reduced the number of times I picked her up. I then didn’t pick her up, but instead rubbed her back and patted it. Then whenever she was quiet, I did nothing except stand by her cot. Gradually I moved away. I thought I would never get out of the room, but over a few weeks (it was not a fast process), I got to the door and then to the other side of the door. If she cries, I go back, pat her back and then go out. The walk in / walk out method is explained somewhere here. Basically I followed the BW guidelines. I read the FAQ about PU / PD and also any other FAQ that I thought were relevant. I was desperate to get it right for her. If she is OT or not wanting to sleep, I still stay with her and pat her back until she is ready for me to leave. It is so much better than it ever was. Now sometimes it will take 25 – 30 mins, but that is rare. Mostly, I can kiss her and leave her. If not, I can just pat her and rub her and then leave her. I think if you withdraw gradually and take a while to do it, he will get used to it. Does he have a lovey? I found that helped my LO a lot. It gives her something to cuddle in her cot. Check whether a lovey is appropriate at your LOs age. I am not sure about that. If it isn’t you could introduce one as soon as you are able. If you sleep with it for a few nights first, it will smell of you.
Is there anyone who could perhaps take your LO out for a walk or something to give you a chance to sleep?
I understand your feelings about giving formula. When our LO was less than two weeks old we were told I had to do that to help her put on weight. I hated the idea, but did it because that was what she needed and I was very stressed so it was what I needed in a way too. It meant I could concentrate on bfeeding her and not worrying about whether she was getting enough because she was topped up. It may help you to relax and get some sleep. Right now, I think you need to focus on that. Getting sleep is so important because it affects all you do.
Anyway, I hope I have not said too much. I am thinking of you and if you need any more advice or help, please let me know. I will follow along. Good luck and I hope you get some sleep.