Chemistry - Atmospheric methane to carbon

Solution 1:

It would be a total waste of energy.

Remember that the Sun is sending orders of magnitude more energy than humanity can produce. A significant fraction of that is UV, so the Sun is already breaking up more Methane than humanity could do by adding laser diodes or whatnot.

We might be able to do more at the ground, where the Sun's UV is mostly filtered.
It would still be an enormous feat to merely reach the Sun's UV level.

However, there is a big reason NOT to do it at the ground: UV does not only break up methane, but also oxygen. Part of that will recombine to ozone, which is an irritant (in the medical sense, i.e. extended-time exposure will cause damage no matter what).
(I do not know whether the oxygen bonds are stronger or weaker than the methane bonds. If methane has the weaker bonds, one could use an UV wavelength below the oxygen energy. You'd still break up bonds of any atmospheric molecule that has weaker bonds than methane, such as pheromones and whatnot; it is entirely unclear what consequences that would have.)

Solution 2:

Powerful ground lasers are surely inefficient. The radicals are created in close proximity, meaning they likely have a good chance to recombine. Also highpower lasers (or any strong light source) are generally not very efficient. A lot of small ones would be better.

I would mount simple UV LEDs on the underside of passenger planes, where they are switched on to illuminate clouds at night (there is no sense in doing it during daytime, obviously). If that's enough to lift the project over the "waste of time and energy" limit, I'm not so sure.