Lava tubes are formed from flowing molten basalt, and basalt shrinks when it cools, forming contraction cracks. If we're producing breathable atmosphere and pumping it into lava tubes, it's destined to leak from innumerable cracks in the cave walls. And it'd presumably leak out faster than we can produce new breathable atmosphere.
This differs from proposed deep "carbon capture" strategies, where we'd pump large amounts of CO2 underground for storage; in that case, the depth provides an airtight seal due to significant overburden, impermeable clays/shales, and perched aquifers above it. But lava tubes, by their very nature, are always very near the surface. That's because lava tubes - unlike other types of caves - form on the surface as molten basalt flows over it. They generally don't survive for long (geologically speaking) nor survive significant sedimentation which would bury them deeply.
Lava tube caves can certainly provide shelter, and can significantly buffer temperature changes. But it would be extremely difficult to use them as an airtight biosphere.
Source: I'm former cave resource manager for the National Park Service at an NPS unit with 700+ lava tube caves which was used by NASA for training and ongoing research.