The NOAA Global Greenhouse Gas Reference Network [1, 2] is a great resource for this. Their data on methane goes back to 1984, so I compared methane and CO2 concentrations in the 1984-2019 time frame. In this time frame, the atmospheric CO2 concentration grew by 19.7% (from 344 to 412 ppm), the concentration of methane grew by 14.7% (from 1630 to 1870 ppb).
So, you're right that CO2 is growing a bit faster, but the difference really is rather small.
In addition to this, there are highly nonlinear effects that may occur in the future that others have alluded to, such as the clathrate gun hypothesis [3]. Even in the absence of such dramatic effects, you would expect methane levels to further increase with increasing meat consumption in countries that are getting wealthier (e.g. in India, China).
Edit: I should also point out that the claim that only future doublings in CO2 concentration matter is totally wrong. Even in the IPCC report's worst case scenario, CO2 concentrations will grow to ca. 1000 ppm by 2100, just over a single further doubling [4]. In the best case scenario, they stabilize a bit under 500 ppm, which is just another 20% increase from today. In that best case scenario (B1), global temperatures are still expected to rise by slightly more than 1.5°C by 2100 [5]!
[1] https://www.esrl.noaa.gov/gmd/ccgg/trends_ch4/
[2] https://www.esrl.noaa.gov/gmd/ccgg/trends/
[3] https://en.wikipedia.org/wiki/Clathrate_gun_hypothesis
[4] https://en.wikipedia.org/wiki/File:Projected_changes_over_th...
[5] https://www.ipcc.ch/site/assets/uploads/2018/02/ar4-wg1-chap... (Fig. 10.4)