That's already possible, the lack of battery, but likely impractical.

There is enough energy during key press/release to be usable for sending radio signal, however it won't be sufficient to do it while holding a key. A combination of a solar panel, piezoelectric keys and a tiny li-ion (as backup) may be sufficient for a 'battery-less' keyboard, but it will be too expensive.

Could you send a separate 'key up' signal on release from the energy of the up-stroke?

That likely would require a (beefer) string to store the energy as release the key alone doesn't require any force