He sounds very similar to where my child was in YR. We now have an EHCP. Having been through it myself here's the info I wish I'd had, hope it is useful if not to you then someone else reading.
It sounds like you need support on three fronts: school, home and diagnosis.
Diagnostic
If you have a household income of under £45k exc benefits, you can get a free ASD assessment for him via Caudwell Children: www.caudwellchildren.com/services/autism/
Failing that, try Elaine the founder of Action for Aspergers, who can do a "pre diagnostic" assessment which you could then use to kickstart the EHCP process www.elainenicholsonmbe.co.uk
You don't need a diagnosis to get SEN support at school (although it helps)
School
Depending on how good the SENCO is you might need to nag. They are usually very overworked. If your DS needs 1:1 for swimming lessons then the chances are he is not fine in school at all - having been told for 18 months that our child was fine in school we have been astonished now the EHCP is in place to discover that actually there have been near daily meltdowns, sensory distress, and even missing episodes. And your DS is not independent with toileting / self care, so he does need additional support.
If you have the funds, fund your own OT, SLT and Ed Psych reports. We couldn't afford all three and did OT and SLT which were the main areas of need. OT sounds like the primary one for your DS.
If not but you can do the 'pre diagnostic' report then that should be enough to establish that your DS MAY HAVE SEN and MAY need an EHCP, which is the legal test for applying for an EHCP needs assessment. IPSEA have a model letter. You can do it yourself if the school can't or won't. www.ipsea.org.uk/asking-for-an-ehc-needs-assessment
At the needs assessment the local authority should send an OT, SLT and Ed Psych to observe your child. Ours was really good, the EP picked up some stuff we hadn't noticed. However I know that some LAs are not great, which is why having your own experts is good if you can afford to do it.
It is a battle but if you have to appeal it is manageable.
Home
Can you work out what is manageable and what is not. e.g. messy eating is something we are just accepting for now, although it's frustrating.
Your DS sounds like he is quite dysregulated. Can you try a visual timetable? They are not just for non-verbal children or those with a LD. A now and next board can also be really helpful.
With the homework (a puppet and a map in YR, those are some major expectations!) I would try to take the stress out of it. He's not going to go from zero to independent overnight even with a visual timetable. Put homework in the timetable for 10 minutes. If in that 10 minutes he draws a line and throws the pencils on the floor then no worries, put it all away and do the next thing on the timetable, ideally something fun.
Does he have somewhere safe to go to when he's dysregulated? You can set up a sensory corner reasonably cheaply, just somewhere he can safely crash where it's not too noisy or bright.
You can get label-free school uniform from M&S in their 'easy dressing' range - also good for speeding up dressing in the morning as it's velcro instead of buttons.
And with the toileting problems have a look at interoception and Kelly Mahler's work to help children with poor interoception.
The lack of support is really, really isolating, especially when you're being told there's no problem and everything's fine at school. Sharp elbows, Facebook / MN support groups, and a supportive partner are valuable equipment.