London clay is very definitely a soil type that moves a lot with variation in moisture content, so needs care with the foundation design to reduce the risk of movement. You can use a raft or slab foundation on clay, but you won't be able to get it that close to the 1m limit, as it will have to have a fairly deep layer of well-drained stone underneath it to spread the load out and down to reduce the soil bearing stress and ultimately limit any movement the clay could induce in the slab.
I think that a piled solution is far and away your best bet, given your space constraints. Piles are pretty much the classic way of getting around London clay heave and would allow you to get the ring beam that supports that outer wall right up to your 1m limit.
You can put an insulated slab on top of a piled ring beam, plus there are several other options that could be explored. I don't think that the suggestion you have at the moment from your SE is at all sensible, given the site limitations. The SE has used what seems to be a fairly standard solution for the soil type, but hasn't taken account of your site restrictions, from the sound of it.
Much as I'm a great fan of insulated slab foundations, there are places where they cannot be easily used, and in your case, with the need to have a vertical load bearing foundation along a 1m line, with nothing going over that line, you are going to find that you're pretty much forced into using piles, I think.
There are lots of piling systems around, ranging from driven steel piles, conventional bored and reinforced concrete poured piles to helical screw piles. The fastest to put in and most probably the ones that would get you closest to your 1m limit, may well be helical screw piles. I looked at using them on a previous plot we were looking to buy and was impressed. It's well-proven technology (some Victorian piers are still standing on ancient cast iron screw piles) and it's a clean and quick solution, with minimal excavation or soil disturbance. It's also a system that is immediately load bearing, so the house can literally be put on the piles as soon as they are driven in.