In my books I advise that most babies of your son's age can sleep through the night, provided all their nutritional needs are being met during the day.
Taking into consideration your baby's weight and the amount of solids he is having, I suspect that most of his night wakings are due to genuine hunger and that you can not realistically expect him to sleep for twelve hours until solids are more established.
On saying this, his feeding three times in the night is probably affecting his appetite for solids during the day, hence the reason he is so fussy about taking solids. When this happens a vicious circle arises of baby needing to feed in the night, because he is not getting enough solids during the day.
Sleep training for this type of problem should never ever be contemplated as your baby is genuinely hungry. What I would suggest is that you gradually reduce the amount of milk that your baby is drinking in the night, which will in turn encourage him to eat more solids during the day.
The first thing I would suggest is to drop the 10pm feed. Because he is waking at 2am and 5am, it is pointless to continue with this feed as it is not helping him sleep a long spell in the night. Consequently it will be most likely that he will wake around midnight and 2am. It is important that when he wakes at this time that you offer him a really good feed and do not restrict the length of time that he is on the breast. A substantial feed should hopefully last him to between 5am and 6am.
As your baby is being fussy about solids I recommend that you work on establishing solids at lunch and tea first. I would advise not giving solids at breakfast until your baby is taking between 5-6 tablespoonfuls of solids at lunch and tea.
If your baby is not having breakfast you may have to bring lunchtime slightly forward if he is showing signs of being hungry before the recommended time. With a baby who is being fussy about lunchtime solids this is preferable to giving solids at breakfast. As your baby is still feeding twice in the night, I would suggest that you offer the solids first at lunchtime, followed by a breastfeed. As he is still taking two milk feeds in the night, you do not need to worry at this stage that his daily milk intake is dropping too low. The aim of this plan is to increase your baby's daytime solids, without leaving him to cry with hunger in the night. You will find that once your baby increases the solids he is eating during the day that he should automatically need to breastfeed less during the night.
Once your baby has been happily taking 5-6 tablespoons of solids at lunchtime and teatime for several days, you can start to introduce small amounts of protein into your baby's food. When your baby is willingly having 5-6 cubes of protein-based meals at lunchtime, then you can introduce breakfast. I would suggest that you start off with a small amount of yoghurt and fruit, then progress onto breakfast cereal and fruit.
The important thing to remember, when introducing breakfast, is that you should not increase it so much that it takes away the appetite for lunch. Also it will really help if your baby has finished all of his milk and solids by 8am. Remember that the aim of solids is to establish your baby on a feeding pattern of three meals a day with bigger gaps between meals. If you give milk at 7/7.30am and then delay breakfast until 8/8.30am it could affect your baby's appetite for lunch.
Once breakfast is introduced I would count the 5/6am breastfeed as his breakfast milk and reduce the amount of milk that he has at 7am, to encourage him to take his solids well. This is obviously difficult to work out if you are breast-feeding, but offering a few minutes less on the breast, then solids, then the remainder of the breast-feed, should help you get the balance right.
Once your baby has increased the amount of solids that he is taking during the day, he should automatically start to sleep a longer spell at night, where you will find him waking somewhere between 3/4am, instead of 2am and 5am. I would continue with giving him a big enough feed at this time, until he sleeps regularly to 7am for at least a week. Once he is doing this you can gradually decrease the length of time his is on the breast by a couple of minutes every few nights. Once you reach a stage of him taking a breastfeed of only a few minutes in the night and sleeping until 7am, then you can look at dropping this feed, confident in the knowledge that the waking is not due to genuine hunger.
I would suggest that dropping this feed would be easier if you can ask your husband to go to the baby when he wakes and settle him with a small drink of cool boiled water and a cuddle. It may take several nights of your husband having to pick him up and resettle him several times. Follow the same procedure until your son shows signs of settling back to sleep quicker in the night. Once this happens your husband should progress to settling your son back to sleep without taking him out of his cot. Although it may take a week or two, by being persistent and consistent you should get your son sleeping through without resorting to leaving him to cry for lengthy periods.
As you decrease the time on the breast in the night, it is important to remember to allow your son more to drink during the day. What you are aiming for now is a full breastfeed at 7am, followed by solids, then lunchtime solids at around 11/12 noon, followed by a breast feed. At this stage he will need a further breastfeed at around 2.30pm, followed by solids and a small breastfeed at 5pm, then a full breastfeed at bedtime. Once your baby is well established on three solid meals a day and four to five good breastfeeds, you can feel confident that any wakings in the night are not due to hunger.
I would also suggest that you have a discussion with your health visitor about the types of foods that you give your baby. Getting the right balance of protein, carbohydrates, vegetables and fruit also play an important role in establishing healthy sleep habits.
Finally, it is important to keep an eye on how much sleep your baby has during the day. Although all babies are different and some will need more than others, I would recommend that you aim for a nap of no more than 30/40 minutes in the morning and two hours after lunch, if you want your son to sleep well at night. If he sleeps less than two hours at lunchtime, then he may need a short catnap between 4pm and 5pm to avoid overtiredness.