Not having "authority" is BS. They have authority.
Authorize their deputies to haul blowtorches to every sprint tower in the county and cut them down. No authority my butt... Just pass a local law stating that towers must be 2 ft lower than whatever height the sprint towers are at, and give them 2 days to comply before the non-compliant towers are removed.
It's called "running the bastard out of town on a rail". Or "tarring and feathering". Gets rid of the unfriendly element in a hurry. If they want to come back, they gotta change their ways or they're not welcome. That's a BIG part of how the constitution was written, to make SURE that local governments have authority over local affairs. If a company fails to cooperate with law enforcement officials in an emergency situation, there is no reason why the local government should not have the authority to deal with the offending company at the local level.
Just cut down their towers, or threaten to cut down the towers, until sprint makes amends. Buying the police dept a new comm system might be a good start at saying "oops sorry". Any other answer is unacceptable IMHO.