Ok, I'm going to tell you a bit about what happened to my daughter and we'll see if there is anything that might help you figure out what is going on here with your DS.
DD actually reacted badly at the time to her jabs, so when she become ill we were able to relate it back to that. She had the five in one and Hep B (we live in France where it used to be routine) on the same day and developed bowel problems very quickly afterwards. Her bowel movements (BM) became very explosive with green mucosy poo shooting out that meant a complete change of clothes every time. She developed terrible colic and wind with constant acid reflux. She cried a lot due to the discomfort, couldn't sleep and started failing to gain weight. She also developed eczema.
To cut a long story short we found out that she was reacting to food proteins passing in my breast milk and having an allergic reaction to them. The worst culprits seemed to be dairy products and wheat but she reacted to many other foods too (all through my milk). She reacted in the same way when she was weaned.
She is now 6 years old and tolerates most foods very well although we avoid too much dairy exposure and a couple of other things that she tests positive for.
We have found a helpful allergist who has explained to me that, in her opinion, the pertussis (whooping cough) element of the DTaP or the five in one can lead to some children becoming sensitised to food proteins. The pertussis toxin affects the gut by increasing intestinal permeability (leaky gut) and proteins that shouldn't, cross the gut barrier, and stimulate an immune reaction. I have read enough on the subject to think that she is right about this theory.
I don't know what is going on with your wee boy but I know it is possible to develop a sensitive gut following vaccination. I am not a doctor and in no way qualified in any of this so just sharing some info mum to mum if you like.
I think the fact that you do not remember you DS having a gastric illness suggests that this could be a vaccine reaction. It is not unusual for doctors to say that it is probably not the vaccine. These types of reactions are probably not very common and many doctors are unaware of them.
I was breast feeding my DD when all this happened so I don't know a lot about changing formula and so on. However if your DS is having digestive problems then you might want to talk to a doctor about a hypo-allergenic milk. Although temporary lactose intolerance does happen, it is more common for babies to react to casein which is the protein in milk (lactose is a sugar). In this case your DS would find a hypo-allergenic milk much easier to digest.
Have you shown a nappy with a BM to a doctor so they can take a look at the white lumps? That sounds to me like undigested milk proteins.
If your DS has suffered a vaccine reaction there are lots of things that you can do to sort this out such as probiotics and digestive enzymes. One very important thing is to try to find a doctor to talk this through with, who will listen properly and take you seriously because if it does turn out to be vaccine related, you will need to be very careful with future vaccines.
I know quite a few people have found Dr Halverson very helpful (he is quite well know for doing single jabs when available).
It is of course perfectly possible that there is nothing going on with your DS other than teething, his milk not suiting or something. Trust your mother's instinct on this though as we know our children better than any doctor can.
Don't hesitate if there is anything you want to ask. You don't say in your post how your DS is sleeping or being generally in himself, how is all that sort of thing?