It's quite simple: burn excess incoming O2 with a bit more carbon.
I don't imagine this would be particularly efficient, however.
GP proposed to filter out O2 by burning carbon. This will create a CO2-rich gas, perhaps at about 30% (depends of the density of CO2 compared to O2).
GP didn't mean that we deplete oxygen in the athmosphere, but that this is an intermediate step inside the CO2 scrubbing apparatus. Perhaps that's why you got downvotes.
The burnt carbon will be gained back by the Gallium process, however this might not work out anyway because the process needs to be very efficient in filtering out CO2, as someone else already pointed out.
In my opinion the idea of burning hydrogen seems a lot better: https://news.ycombinator.com/item?id=29994000
I do wonder if it would be much cheaper to capture CO2 by investing technology and capital into 'building' forests. What if we can irrigate Sahara and cultivate some fast-growing plants like bamboos there? Maybe that has ecological consequences but it doesn't seem to be much different than the agricultural land expansion that has happened in rainforests in the first place.
However, we do know of how the deposits of fossil fuels formed in the first place, and most places newer than the carboniferous, formed at the bottom of deep lakes and especially oceans. They form from the anoxic compaction of marine snow. Either during massive algae booms, or more slowly from slightly larger corpses. This process is slow for several reasons, but a significant, in particular for the latter category, one is the lack of surfaces for attachment on the open ocean. The creation of large floating artificial coral reefs could provide this, and with it a highly productive rich ecosystem. We could further fertilize these, with iron in particular being rare, but also regular fertilizer.