Both Andy's and Aubrey's observations are most interesting. The key observation being the dryness of some of the unenterable passages. In Halloween Rift, where we have been following the main draft, the sediments and speleothems are extremely dessicated. Just touching the silt/clay sediments at the top of the pot results in an avalanche of sediment similar to that in an hourglass. Although that cave has only one entrance.
A plausible explanation in this case, may be some sort of venturi affect, similar to how a paint spray gun works. If the main volume of air flow is from a lower unenterable passage in Rod's to Rod's entrance, this would provide a suction effect to the passages that link with Bath, creating a vacuum in Bath that causes the inward draught. Its the only explanation I can think of. Dryness may indicate the main way on?.
Just a few thoughts that may help?