Why are $\small\sf{CH_4}$ and $\small\sf{CO_2}$ levels coupled during the Pleistocene?
This question is addressed in a Nature paper on "Understanding the glacial methane cycle" (Hopcroft et al. 2017).

There conclusion is that changes in methane concentrations are driven by changes in the biosphere associated with the global temperature changes. The dominant factor is the reduction in methane emissions from wetlands in a colder climate. There is also a reduction in emissions associated with forest fires in the cold phases (colder climate means less forest fires; warmer climate means more forest fires).

Note that the model used by Hopcroft et al. has the CO2 levels prescribed. In this situation the temperature changes in response to the CO2, and then the methane changes in response to the temperature.