I found with Olivia, she hated wet diapers. in 24 hours we could go through 16 diapers - but she stopped crying shortly after diaper was changed, thus wouldn't work herself up & be constantly overtired which we all know is no fun at all. Now that I think about it, maybe the motion of lifting her legs helped her pass gas too. I'd put a drop or two of oval in her formula & this helped tremendously with the trapped gas issue as well. How about having the bottle ready to go for the 5:30/6am feed- in his darkened room, sufficient quiet burp time (15min), diaper change (we had to use caution with Olivia, milk would come up if layed down too soon after feed), them put him straight to bed again, start your(his) day at the next wake.
Check that he is not Lactose intolerant, Olivia did this as well, switched milk, things better again. HTH
I found having a little crock pot on low temp all night with about 8 ounces of water in it night helped when she woke. I'd have her bottles in a cooler in her room during the night/days. Upon waking, I'd put a bottle in the crock pot, change diaper & viola! 4 oz milk perfect temp. no going up & down stairs, warming kettle, making milk, overstimulating baby in the process.
p.s. crock pot omitted dry air in her room too...just remember to turn it off in the am & top up/change water every night. FYI, takes 3 days for 8oz of water to evaporate in a crock pot, so I felt pretty safe doing this...