Two children of similar age closely related to me sleep on a mezzanine, approached by a spiral staircase. It has not been a problem. But the rails on the actual mezzanine are bigger and stronger than those in your pics.
There are some problems with the design at your house. It needs a rail on other side of the stairs and all the rails need to be of a sturdier construction. Also, there needs to be a proper space between the bottom of the stairs and the wall/window. I am surprised that your builder was happy to put it there like that with no floor space at the bottom. If these changes were made it would be safer. I suspect that the building regs person will suggest these and other changes.
Some boys would think it was wacky to sleep on a mezzanine, as indeed my relatives do. This is why children love climbing up to zip wires and on climbing frames. BUT........the construction needs to be safe and they need to have choice, which is sadly lacking here.
It does not sound as though your OH understands what compromise is about, as I am sure there is a compromise here. As long as you and he do not agree about this, then the children are going to feel they are being manipulated, as indeed one of your DC has virtually said.
I think you need to try and separate out the fight between you and your OH and the matter of the boys wanting/not wanting to sleep up there. At the moment they can please neither parent and they must be feeling quite uncomfortable. There is a bullying element to your OH's approach to this, which I am sure betrays an underlying problem in your relationship. But please do not let the boys be pulled from pillar to post - get this argument done outside of their hearing.
I can see that he must have thought he was doing something that the boys would love - maybe it is just the sort of thing he would have loved - but it has become a big bone of contention now; and your Mum has been drawn into it too.
Would it help at all if you tried to say to him that you understand why he thought it would be a good fun option, and that you are happy to go along with it if proper safety measures are in place and the boys can make their own choices about when/if they sleep up there? This would be the most sensible outcome.
This has become a focus of your marital discord and the boys are piggies in the middle. If you do not like this man and want to end the relationship then so be it - but the boys need to be detached from this dispute rather than being the pawns.
He clearly thinks you are making a fuss and is exasperated that what he saw as a fun project that the boys would love has now gone belly-up , and the visit from building regs should settle that. But the remaining tension and discord is not going to go away easily.
In a sense you are involving Mumsnet in your fight, in the same way that your boys are involved. I do not think that is good for them.
Sorry if I am not saying exactly what you want to hear.