Lots of infants and children with food intolerances are poor sleepers, ( low weight gain is linked to that as well sometimes)
A leaky gut can take a long time to heal, mainly because it cant rest, as we need to eat!
Often when layed out flat in bed the gut stretches and causes pain. which is one reason why they dont sleep.
Normal babies learn to sleep for a longer time as the weeks go by, a healthy newborn will sleep 4-5 solid hours.
Then by 6 months most are sleeping longer.
Some allergic and intolerant infants have never been able to sleep for this long.
ezcema can cause poor sleep, as well as gut , and tummy pain.
My son was unable to sleep for long periods due to his allergies and eczema, he finally slept for the first time a solid 4 hour sleep at 9 months.
He was developing a sleep pattern at 9 months, like a newborn baby , and carried on like that until the magic night when he slept through.
It also coincided with the discovery and removal of his food allergies, and environmental triggers.
Sleep training was attempted , promted by health visitors, but was not a wise choice for him.
leaving a child to cry for even a short time when they have ezcema , only leads to a lot of blood, skin flare up, and wet cotton wool needed to soak away cot sheet from skin.
I wouldnt rule out the sleep training, as it worked for my normal healthy first born, but would consider carefully all the options you have.
I certainly would not jump to stopping breast feeding. It may be that your child has other food intolerances, but without a normal weight gain and sleep pattern, at least your child is protected against infection via your milk.
If he has a cows milk intolerance, there is no harm in breast feeding until 2 yrs or more.
introduce new foods one at a time, and if at first refused try again once the following week or so.
This isnt a race, he is still feeding and will eventually increase solids, he is on his own time scale.
He may also be picky about his food for very good reasons.
only time will tell.
does baby sleep for a short period during the day?, when he does, sit down. do nothing.
in fact , sit down at first flutter of eyelids, its your cue to rest.
if you dont have older children, lay down, get a nap.
push all rubbish under the sofa, it will keep for another day.
eat well yourself, dont miss meals.
good luck.