I have a 4 1/2 month old and have similar issues as you. I think breastfed babies just want to be close to their mamas! :-) Anyway, here's my rough schedule (because I feed on demand too and I only ever do one side because I have SO much milk if I do both sides, his diapers reflect too much foremilk).
6 am nurse
10 am nurse and down for 20 min nap
12 pm nurse and down for 30 min nap
4 pm nurse and down for 20 min nap
8 or 9 pm nurse and falls asleep for night
2 or 3 am nurse and sleeps in bed with me for the rest of the night (if I try to put him back in his pack-n-play/cosleeper by our bed, he will wake up in about an hour to nurse again until I just let him stay in our bed).
My kiddo is funny because we both fall asleep when he's nursing, but sometimes I will suddenly feel this little mouth searching (not awake) and he will relatch himself, take a few swallows and then stop again! This is my 2nd baby and he needs to be sucking on something so much more than his brother did (my first loved nursing, but in between times, he didn't need to have something in his mouth...he only sucked his first finger four about a month and never took a paci). I have tried doing the paci at night instead of just letting him nurse again beyond the needed 2 am feeding, which sometimes works and sometimes does not. During the day, he sucks his first 2 fingers, but it is like he forgets about his fingers at night!!!
My only suggestion for you (because everything sounds pretty typical for a breastfed baby) is that maybe feed the baby once more right before you go to bed and maybe that will eliminate one of the other nighttime feedings for you! I have to say, I love snuggling with my little one in bed though as I FINALLY got my cuddle baby (brother is Mr. Independent!)!