If I recall correctly... my very first post on Reddit was doing calculations for a (practically immortal) person eating beans and storing the flatus for a trip to the moon (searching shows that this is a not-infrequent request). It was only concerned with quantity - not storage or the engine.
... and the source document for the numbers was based on a paper that is fairly easy to find given the proper keywords in google search... https://pubmed.ncbi.nlm.nih.gov/1648028/ (and I learned that methane more rare in flatus than not).
Did you account for the weight of storage? Because I would think the tyranny of the rocket equation for such a low ISP would cause various levels of impossibility, not to mention the problem of getting enough thrust.
I think the scale was on the hundreds of thousands of years. We're dealing with 700 ml of hydrogen and 70 ml of methane at standard pressures and scaling this up to 90,000 kg of hydrogen and 635,000 kg of kerosene (with the 1:1 methane).
90% of the atmosphere is below 16 km.
16 km * (12" / Earth diameter) :: https://www.wolframalpha.com/input?i=16+km+*+%2812%22+%2F+Ea...
0.015 inches, 0.38 mm
... and tossing sheets of paper into that ( https://www.wolframalpha.com/input?i=thickness+of+paper ) ...
16 km * (12" / Earth diameter) / thickness of paper :: https://www.wolframalpha.com/input?i=16+km+*+%2812%22+%2F+Ea...
4
Note that that's copy paper rather than card stock...
Adjusting this to 5.6km (the 50% atmosphere amount) ...
5.6 km * (12" / Earth diameter) / thickness of paper :: https://www.wolframalpha.com/input?i=5.6+km+*+%2812%22+%2F+E...
1
So it's a matter of selecting the proper globe, proper paper, and proper threshold for the atmosphere.