The methane cycle I'm familiar with is related to things dying and getting buried underground or in the ocean. I don't think there is any comparable process to how photosynthesis consumes CO2. There are methanogen bacteria who consume it, but that's probably not a globally scalable solution. I think it would be cheaper just to capture and store it.
I know that data for CO2 concentrations in the atmosphere go back at least 541m years, i.e. the whole Phanerozoic. Does a similar dataset exist for methane?
https://en.wikipedia.org/wiki/Hydroxyl_radical
https://www.sciencedirect.com/science/article/pii/S092575351...
I wonder what the feasibility is in seeding the troposphere with hydroxyl radicals to accelerate the breakdown of methane?