The rule has been clear for many years that an individual citizen does not have legal standing to go to court to question the 501(c)(3) status of a charity.  Only the Internal Revenue Service can question or revoke such status.  You can file a complaint with the IRS, but even if the Service did conduct some sort of investigation, it is unlikely that the IRS would strip (c)(3) status because of a disagreement over the quality of care without a showing that it was totally outrageous, perhaps criminal.

You might have more impact in changing conduct by complaining to the state or local authorities in charge of oversight of the shelter.  They are much more likely to conduct an on-scene investigation to enforce local rules.