My little boy (now 2) was the same and I hope that by sharing our experience it can help you or that you can pick out something that sounds like you and identify what's wrong.
Firstly, my LB drank a lot of milk (he wouldn't have stopped by himself, just got in more and more pain) but he fussed during and screamed after feeding, so we knew he had silent reflux. For him it was a combination of tongue tie and cow milk protein allergy (lactose free milk won't help this). This caused a LOT of crying but that wasn't the main problem...
Secondly, my LB like your LO was almost impossible to get napping and that (I only realised at 6 weeks) was the main cause of his constant, inconsolable crying. He was massively, massively shattered. I worked very, very hard on establishing good nap habits and I won't lie, it took a long time to truly get there, but I saw light at the end of the tunnel very quickly after starting. A website called Troublesome Tots helped me to get some ideas, but it turned out he would get tired around 45 mins after waking up. We had to use every trick in the book to get a nap and we even had to vary it through the day because he would get wise to us. We would get 30 mins at very most, but wow, what a difference that made! It was as if he had to learn how to do it. Like your LO he always was (and still is) good at night and sorting out day naps made him even better at night.
Lastly, my LB hated cuddles (this made the whole sleep thing above much harder) and was very sensitive to sounds and textures. He was frequently bored but also very easily over stimulated. He is a highly sensitive child and we are querying sensory processing disorder (he won't let us put TV on at the moment because it's too noisy). I'm sure this added to the sleep problems but it also led to quite a few over stimulation meltdowns (which was more obvious from around 6 months).
I don't know whether any of this us relevant to you but I truly hope something in there will ring a bell. I ended up with PND too and I honestly believed that he hated me. Please Google 'Wonder Weeks' as it helps to explain why babies suddenly get much easier at certain ages. My LB is still very sensitive but he's a very well-behaved and clever little boy who seems a far cry from the angry little thing who I felt like I wasn't bonding with and was generally failing for the first few weeks of his life. I send massive hugs because it's such an awful place to be, but I promise that the changes are rapid and positive. Xxx