Just looking over that wikipedia link, I'm seeing a lot of references to British monarchal eras and Highland fashion. So it could be that this was an item primarily prevalent in England and Scottland.
England is not exactly famous for its sunny climate, and average temperatures there even in the summer appear to be a few degrees cooler than in Poland (where you are performing this re-enactment?). Where I come from in the middle of the Continental USA, it would be considered downright cold. We regularly have days a good 40 or 50 degrees F hotter.
Scottland is even colder, murkier, and damper. A wool lining would be highly desirable there, both for its insulating properties, and for its wet-weather behavior.
Also that area was pretty much the wool-production capitol of Europe. So the material would have been much more plentiful (and cheap) than it would be most other places. Most Scottish nobles could probably skim it for free from their own estates.
So I don't personally find this all that unlikely. However, garb in Poland may have been different, and summer garb in Spain or Greece was almost certainly different.
For my money, unless you can import Scottland's climate for your re-enactments too (perhaps run them in the spring or fall?), I wouldn't take versimlitude to quite that extreme. If you are sweating like a pig due to overheating in a non-historically accurate climate, that's a bad re-enactment too, isn't it? Then again, I'm not much of a SCA or LARPer, so I don't know the culture like you do.