I've read the whole thread now and have been keeping with your posts.
Good luck with all the referrals etc, but I think you can see yourself that it may not be SN but the relationships with you, and with his sisters that is the issue here.
When you're stressed out and exhausted it's hard to be doggedly consistent but that's what he probably needs. Kids need to know what they can expect from you, and in this case it seems that needs to be that they are all equally loved, and will all be equally treated.
When he drops the jigsaw, check first that your baby is ok and put her in a safe place. Then go straight back to him. His speech is not great and maybe his understanding is not either, but he will understand a very sad face. Be calm and speak slowly. "That was a wrong thing to do. I love you very much. I love your sister exactly the same. I cannot let you hurt your sister. It makes her sad. It makes me sad. You must not do that again. Let's be gentle with xxx and play xxx" If he gets cross or sad or screams, that's fine, it's only an emotion - he can't handle his emotions yet so he needs to learn from you and he needs to see that you CAN handle his emotions, so he doesn't need to be scared. You might like you're repeating yourself 7,8,9 times - it doesn't matter. Sometimes it takes a week of me saying 'we do it like this, not like this...' or whatever, before it sinks in.
The big thing I think is that you need to make him feel that you love him every single day not only when his sisters are absent but when they are there. As his speech gets better, if he hurts them you can ask him to say sorry. So when they are all together say 'I love you all exactly the same way'. Do you know that book - 'You're all my favourites' - if you can hold of a copy from the library try that...
www.amazon.co.uk/dp/B00APV169S/ref=dp-kindle-redirect?_encoding=UTF8&btkr=1&tag=mumsnetforum-21
When we moved flat (we're in a 2 bed as well) I read my 4 year old son a book every night about a little tiger who moved house - for about 2 weeks before we moved and two months after...the message really sunk in...that home is where your family is, not where your house is. What needs to sink in for your son is that you love him ALL the time, on his own and with his sisters, and that everyone will feel the love better if you are all gentle with each other. (If he grabs the book don't let him. 'No. We don't grab books. Mummy is holding it. But if you can hold it gently, we can hold it together.' If he kicks off 'You're feeling cross, that's ok, we'll read the story later.'
I found it SO helpful to help my son NAME his emotions. 'You're feeling very angry right now....What is making you angry? If he can't talk, sometimes I say, 'Shall we open the door and swoosh the angry away....' (I know this sounds bonkers but that worked when we had some quite bad tantrums).
I say all this knowing that it is almost impossibly hard to be as calm and loving as you need when you're so tired! But I've noticed the difference since I gave my son the ability to know that I'm cross or sad with something he has done (his behaviour) rather than that my feelings towards him as a person have changed.
One idea: this sounds a bit mad, but sometimes in the heat of the moment when your emotions are raised and you're panicking the only thing you can access are things that you're rehearsed or practised.
Try saying them at night when you're on your own or when he's at nursery. Practice in your head gathering them together, and saying a firm but calm 'No, I can't and won't let you hurt your sister/s. Let's do xxx instead together...I love all my babies (include him in that) so so much.' If he objects to being called a baby, great 'ok, I love my babies and my big boy so so much...and you're going to be such a lovely big brother...'
Honestly I know it sounds mad to practice it but when you're exhausted and angry and upset it's hard to remember anything except the things that you've drummed into yourself.
You also might need to get some headspace / sleep and prioritise doing something nice for yourself with a friend or something to remind yourself that you're a strong mum and his behaviour is not beyond your ability to cope.
(The more he sees you unable to cope the more out of control he will feel and the more he will act out).
You've had a bit of stick from people who say you've just come on to moan. I don't see that. I think there are things you need to say, to yourself, to someone else, so that you're not saying them to your son, and you can sort out what you're thinking and feeling. I think you're brave for sticking with the thread to be honest. Once you've acknowledged your feelings you can just let them go. Same with your son. You can't tell him what to feel, he feels what he feels, same as you, but you can both start to change behaviours, and the environment.
This book is also really good for showing how children start to live 'down' to expectations but can also live 'up' to expectations if adults start from the point of view that there are no bad kids....only kids who do good and bad things...
'Edwardo - the horriblest boy in the whole wide world'
www.amazon.co.uk/Edwardo-Horriblest-Whole-Wide-World-ebook/dp/B0060MGHWM/ref=sr_1_6?s=digital-text&keywords=john+burningham&tag=mumsnetforum-21&ie=UTF8&qid=1548151052&sr=1-6
If you read that with him (it's in our local library). Ask him how he feels about Edwardo, and the grownups in it then tell him he is a kind boy, and can be a lovely brother - remember when you helped xxx, remember when you got cross but then you said sorry...remember when you walked so nicely by the pram?
That sort of stuff works but you need to 'fill up their little buckets' with it every day xxxx
Good luck!!