Hmm, well that's a little more than you posted again. Do bear in mind though I am not a legal professional or legally trained but I have experience of this (similar situation) as a mum with an ex who I share residency with and as a step mum who's partner shares residency with his ex. Plus I have done a fair bit of my own research (and lurking on threads on here for far too long of course).
The Ofsted report won't be taken into consideration, any school that is open is deemed satisfactory or at least under special measures to improve. So a better Ofsted report isn't a good enough reason to move him.
Sleeping on a mattress on the floor isn't ideal, but hardly a huge problem. Plenty of people live in a 1 bed place with a child or two. (I did years ago) I don't think this would win you an argument at Court as his dad will just offer to give him the bedroom and they have the living room as their bedroom (he will be advised to offer this by his solicitor of course). Plus you have been happy with him living there 3 days a week for the past year AND you propose for his dad to still have him there 3 days a week even if you move his school, so you can't say you're dead against it, 3 nights or 4 nights a week isn't a huge difference and sort of dampens your argument.
The fact his dad has a childminder collect him from school every day you you are available everyday may be useful, but not the be all and end all. His childminder 3 days a week is a routine for him and he will likely have friends there and get to play with school friends there too. His stepmother's mum collecting him will be seen as him spending time with his extended family, so not an issue really. Remember your ex could counter argue this and say IF he had him school days he could apply for flexi-working or cut his hours down and be available for most school pick ups too, or his new wife may offer to pitch in (I would do that if it were me at least) Don't forget if he has him school days he would be the "main" carer, he will be able to claim the Child Benefit and therefore claim maintenance from you, with that he may be able to afford to work a bit less and be there a bit more. Not that I know what he's like, just possible things that he may say in Court.
A "good" reason for changing schools?
The current school isn't meeting his specific needs ie he has special needs that can be better catered for at another school. This would be a very valid argument.
You seem really sure you want this to happen though so why not just make the application and try? You need to fill out a C100, you can print this form online if you Google it, take 3 copies (and the fee of £200) to the Court and they will book a hearing for you for the near future. You will need to state all your reasons for wanting him to move schools on the application, all the benefits to you son as arguments for moving and also how you propose the relationship between him and his dad is not damaged / contact limited.
I do think you need to serious consider that if you and your ex disagree the Judge may make an order that nether of you are happy with. The Judge may decide it is better for your son to stay at his current school and or your ex to have weekdays as the journey (you've said yourself) is too long for him in the mornings. It's a roll of the dice if you decide to make the application really, it's up to you if it's worth the risk. Personally, I'd just move back.
Without wishing to Judge or insult you i think you made a huge mistake moving away before you had this issue sorted & you'd have stood a greater chance of winning at Court if you had made an application to move his school before you moved house. (Plus then you wouldn't be stuck 40miles from your son's school).