Is a public school district considered a government entity for BSA exemption purposes? If so, is it a mandatory exemption and we don't have to wait until they hit the threshold for transactions?

Phase I exemptions are exempt based on who they are. If the entity is receiving property tax income, it is a government entity and is exempt immediately.

(Note ansicllary entities associated with the school district likely are not government entities such as the local PTA chapter, scout troops, alumni clubs, sports and music boosters, etc..)

Phase II exemptions require five reportable transactions in 12 months to qualify as an exemption.

A public school is usually a subdivision of the local government, which would make them a Phase I entity and not subject to any transaction threshold tests.

"District" implies that this entity operates across county lines and is not a subdivision of any one county government. If that's the case, state law should spell out what it takes to be this special type of subdivision of the state, not of a single county. If the laws of the state or ordinances of the counties served by the District authorize the District to levy fees and charges to fund its operations, these should probably be viewed as the equivalent of taxes.