He seems to be a really weird interesting mix of being both over and under sensitive in different areas.
That's absolutely common.
It puzzles me that school don't see the behaviour that we see
That's really common as well, look up 'masking' what happens is the child is academically okay so school aren't concerned. Behaviourally the child's used to school routines so just jogs along and doesn't act up there in any noticeable way. Many children,mine included could have been helped years earlier but because they held it together at school, school denied there was anything wrong and blamed poor/helicopter parenting.
BUT, and this is the BUT that school don't see and rarely acknowledge, all day the child is like a bottle of fizzy pop and everything that irritates them is like shaking the bottle once, then once more for every irritation. At hometime, the child knows home and safety is near and sometimes coming out of school or coming into the house is a point where they can't hold it in any longer and they have to explode because that's where they are loved and feel safe enough to do it.
He's young yet, but the NHS diagnostic process is very slow, an assessment for example for autism takes between 18 months to 2 years, some other conditions aren't even assessed until a child is older, so keep an eye on him and have an open mind and push for an assessment if you think he's very different to his peers.
Most schools although they should provide interventions based on a child's needs tend not to do so due to funding and a diagnosis can in some instances unlock that funding.
If he's having sensory issues that disrupt his day, then try and refer to a Paediatric Sensory OT Occupational therapist. The NHS used to have some, check in your area, if not you may find a few private sessions identify his needs and suggest interventions.
The whole question of having a child assessed should be based on how much that child is affected by different things and in turn how much of that affects their daily life.
You will find that your concern will be minimised by friends family teachers and often the GP, "all kids his age do that" but if you feel strongly enough that his behaviour is definitely different to his peers to a very strong degree, then do take a list of all his differences to your GP or see a different one and ask for a paediatric assessment. As school aren't helpful right now because they "don't see" anything, if SENCO is approachable you could ask her opinion, if not, you'll just have to be firm but persistent with the GP and state which things affect him to a much greater degree than his peers.
Also, many kids with additional needs appear emotionally a lot younger than their chronological age, two thirds is a rough rule-of-thumb, so you may find school notices that.
Hang in there, keep observing, keep reading about different conditions and see if they apply to your son in a major way, then you can decide whether to push for an assessment. We all have traits of many conditions, say a sensory one is irritated by seams in socks, if someone's irritated for a few minutes then forgets, it's not noteworthy, but if someone can't bear the feel of it to the extent that they feel it's like having their toes sawn off and they have to remove their socks or they'll scream the place down, then yeah, that's a strong reaction and definitely needs noting to tell the paed, OT or whoever.