OP, this sounds truly shit, for both you and him.
I see you have considered the main food intolerances, but it is possible he has multiple intolerances, to more than the usual culprits, and that is why cutting out the big four didn't have much of an effect. To give you an idea, my DD has CMPI, but is also intolerant of gluten, soy and a whole host of usually innocuous fruit and veg, including grape, date, spices like cinnamon, and previously stuff like carrot and banana (though she can tolerate these now at just turned 3).
You could request a trial of a hypo-allergenic formula like Neocate or Alfamino, to see if symptoms subside (or buy a couple of tins, though it is £££). The problem is, that he is almost at weaning age, so you might get pushback from the GP, but it would give you a chance to see what he is like when all obvious allergens are removed from his diet. Introducing solids could confuse the picture more, if intolerances are in play (though of course, they might not be....).
It would mean an end to breastfeeding, though, as you would never be able to cut every possible source of intolerance out of your diet. He doesn't seem to be happy with breastfeeding, mind, and like a pp, I know a couple of babies who were constant whingers while ebf, who became much more content and satisfied once formula was added.
Tongue tie is another thing to check, as pp suggests. It can make for inefficient feeding, leaving the baby permanently hungry, and constantly in pain from wind. Worth a check with someone who really knows what they are looking for.
When it comes to weaning, don't necessarily wait until he can sit unaided, as some babies are not doing that until 8 or 9 months (though you have time to trial a hypoallergenic formula first, if you get your skates on). The Ikea Antilop can be bought with a cushion that you place in the seat to make it snug, and hold the baby in a sitting position. It is also handy to take to the park to put in the baby swings, for the same purpose.
Quite honestly, I would be demanding a referral to a paediatrician by now ( I can recommend a good one if you are in London?). This is not normal, something is making him unhappy, and chances are it can be sorted, if only it can be identified.