'> some light but springy wire.
You can often get 'piano wire' or 'music wire', which would do fine."
Colin used superelastic" nitinol (nickel-titanium alloy) wire. This is commonly called Memory Metal. This wire is manufactured into straight wire which sets the memory. You can bend straight wire into very tight loops and the wire will spring back straight. Nitinol will easily outperform Piano wire or music wire The superelastic alloy is I believe 56% Nickel and 44% Titanium.
"Can you add a spring to the screen to make it pop up if the stiffness degrades?"
Just stich two vertical nitinol wires between the nitinol hoops. If the fabric softens the vertical nitinol wires combined with the nitinol hoops would maintain the shape.

