The shed-hunting has turned into a family affair on Aspö. Ann brought over a magazine with a number of possibilities. A new company they found was Jadestugan, but it appears they make the same type of 'plywood' sheds of which Janne disapproves.
He and Margaret spotted a potential alternative close to home, however. A new house is going up on the path near their (and Ann& Dave's) home, and a little stuga was built there a few years back as an initial building. It appears to be a step up from the thin sheds I've been looking at, and it also has the same slanted-roof profile of my X-Houses.
Janne talked to his neighbor and found they were made by yet another prefab provider, Caboden. I checked their website and saw the "Mini Funkis" range, which does indeed look like my X-Houses. As best as I can tell, they have a 6m² model for around 18,000kr, which is double the cost of the cheaper sheds, but half the cost of a full X-House. I take that as good news, as it's likely to be much more sturdy, yet it's still (probably) within my budget. This might be just what we were looking for!
I have sent an email to the company asking if I could talk to someone in English and I'll report back when I (if?) hear from them.