Maybe there's a solution!
What if, whenever there was a fire alarm activation that could not be readily investigated, the care home staff called a private fire fighting company. The private company wouldn't need all the resources of the fire service, but would have just enough to ensure an initial investigation and fire fighting response. If they attended and found that additional resource was necessary, they could make a 999 call to the FRS and then withdraw upon their arrival.
The private fire service would deal with all those nasty unwanted fire calls, and would charge the care home appropriately for their services. This would leave the FRS to respond to real emergencies and do the fire safety things they do.
There is a potential downside though, the number of calls sent to the FRS may reduce to the level whereby the need for local resources may be questioned with the result that the local fire station loses appliances or changes to retained. This may mean that the availability of FRS resources in response to a 999 call by the private company may not be assured resulting in the care home provider having to consider the potential of the greater risk. Perhaps the private fire fighting company could provide sufficient additional resources to deal with the problem.
Hmmmmmm, seems that the fire service has somehow ended up privatised!!!
This is all nonsense, but so too is refusing to respond to a sleeping risk involving vulnerable occupants!