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Not an HVAC tech, but where are your air returns located? I would agree that keeping the fan on will help mix the air in the house. Closing all the registers on the lower floor might stress the system, but you could close a few I’d wager and be safe.
Bees collect nectar for nutrition and they collect pollen as their protein source. These activities pollinate as a by-product. Honey bees collect a lot of nectar to refine into honey to survive the winter months. This makes the honey bee ideal for some flowering plants which have a lot of nectar that the honey bee needs. Many native species have short lifecycles. Some only during a bloom of a particular flower. This is why native bees make better pollinators as a whole. Honey bees pass up many flowers that have little benefit to them as a nectar source. Native bees collect the pollen that would be undisturbed by the honey bee.