Maybe, but large organizations like UPMC, whether they call themselves 'non-profit' or not, are driven by profit. Diverting police away from the hospitals is likely just going to create a public relations nightmare.

The effective way to pressure UPMC is to impact their bottom line. Use the permitting process to prevent their new hospital wing or administrative building from opening for even a couple months and they will lose millions of dollars. And that is money they will lose immediately, before they can effectively retaliate monetarily by forcing the city to pay legal fees fighting in court.