It is extremely common for things to get worse, at least temporarily, when children first go into the juniors. It was the first year of the juniors when things really came to a head for ds1 and that's how we ended up with a proper assessment and diagnosis (AS). Similar case for us, the school swore they hadn't seen any red flags prior to that, despite us raising it with them many, many times.
So much changes in year 3. The expectations of childrens' behaviour, the classroom set up, more rigid timetable with less free-choice/play, less classroom support as less TA's allocated to junior classes, one less playtime and on top of all that the social side of things starts to really change as their friends mature and start leaving them behind developmentally. I honestly think years 3 and 6 are the worst possible years for children with ASD. I was beside myself with worry about ds in year 3 and we are right back there again now he's in year 6. 
Ds has similar sleep issues as well, again, worsening when he entered the juniors. He's 10 now and sleep is one of the first things to go haywire when he's stressed.
Things that have worked for us are:
- Having a talking/download session before he goes to sleep so that he can release any worries before he tries to sleep.
- Making sure he has a good hour of quiet time before bed, so either reading with us or on his own now he's older or listening to some calming music.
- Having a set bedtime routine that includes things which reduce his anxiety. For my ds this includes saying a little prayer and having the same song sung to him, as well as allowing him some degree of ritual regarding the positioning of things around his bed eg his favourite cuddly toy, his bedside clock and his little angel figurine. (I know it sounds like we are religious, but we're not, ds just finds these things comforting.)
I have read some good things about some of the child meditation cds that are available on Amazon and want to try some of those sometime soon, although apparently they tend to either work brilliantly or make things worse depending on the child. There are a couple on there that have positive reviews written by parents of children with ASD, so we are going to give those a go.
Another thing his ASD teacher has suggested is to get a little notebook, sit quietly with him each evening and have a chat, helping him to come up with at least three nice things about his day. They don't have to be big things, perhaps just he laughed at something on the television, he enjoyed his sandwiches etc. Apparently they call these 'Sparkle Books' with younger children, but my ds just wants to call it his positive thoughts book. I think the idea is that it gets them to focus on happy thoughts and this helps to relax them before they start trying to sleep. I have no idea if it will work, as we haven't really got into it with ds yet, but it might be worth a try.
Finally, not sure if this will be of any interest either, but my ds's sleep pattern definitely improved when we started him on high strength omega oil. We just add it to his porridge once it's cooled in the morning or add it to yoghurt, as the one we have is vegetarian so doesn't taste really fishy like some do. If you do a quick search on MNSN, there are lots of threads with advice on the best omega oil pills different people have tried.
The paed at ds's Statutory Assessment suggested melatonin, but at the time his sleep wasn't too bad, mainly because he was happy and well supported at school. Fast forward a year and I wish we'd taken her up on it, because he is literally not sleeping at all at the moment and we are all exhausted as a result.
Ultimately though, the only thing that is really going to help is making sure he is properly supported in school, to reduce his stress and anxiety levels. You say someone has suggested he may have ASD to you. Has he been referred for assessment at all? If the school are unwilling you can go to your GP and ask to be referred, preferably to a developmental paediatrician.