or you could increase the size of the wings?
I think you got this upside-down. The Kármán line is named after an engineer called Theodore von Kármán who studied the problem of how high aeronautical flight is possible theoretically. That has everything to do with lift, and speed an orbiting body needs to stay in orbit. That is the Kármán line.
Then yes of course organisations need clear and bright lines for things like "when do we give someone an astronaut badge?" For that different organisations come up with nice round threshold numbers. And those numbers have nothing to do with lift. But those numbers are not the Kármán line. They are just convenient approximations of it for administrative purposes.
It's like if a country's tax authority declares that for tax purposes you shall calculate the area of a circle using the first 5 digits of PI. That doesn't mean that suddenly the PI become not a transcendental number. It is just an approximation for practical reasons.
They aren't, though. They still work, just much less well. There isn't one altitude where wings suddenly stop working entirely - there is however an altitude where generating enough lift to sustain flight would require going so fast that you're already in orbit (regardless of lift), according to wikipedia it's somewhere around 80km.
The Karman line has nothing to do with lift, it's just an arbitrary, round number.