3M Very High Bond (VHB) double-sided tape, available from a trade bodyshop/paint supplier, should do the trick. Usually has a RED backing film - don't waste time with the green backing film double-sided tape, it's pretty weak.
I wonder if you can pop down to Airbus in France and pick up some VHB tape from them - it's used to hold parts of the wing together
And windows on skyscrapers, etc. I don't see leading-edge slats falling off Airbus aircraft or windows falling off the Eureka tower, so you'd think VHB tape might just hold the skirts onto a 156 for a while.
Surfaces really do need to be perfectly clean and smooth for new tape to work. Cleaning off the original adhesive remains is likely to be quite a pain. Methyl Ethyl Ketone (MEK) softens it so you can remove with a scraper, then rub with MEK again. No solvent will just remove the adhesive in one hit - it's the most tenacious adhesive known - but the approach seems to be soften with solvent, scrape with plastic scraper, then buff off with solvent again.
Otherwise, I think people in England have a problem with leaking tigers, because there's a sealant sold for them. We don't have tigers here (most people have smaller cats) so I haven't tried Tigerseal, but I understand that works well for sticking bits back on cars
The Autoglass polyurethane for bonding windscreens into cars sounds like a very good solution. My only concern is that I thought that required heat curing (through resistive wires or through infrared light), which won't be practical with plastic parts.