As it stands, the U.S. court system is barred from hearing cases that are not based in some factual question and the person petitioning the court can be potentially harmed by the law in question (standing). Thus we cannot say for certain how the real life court would rule as demons have made no such claim.
That said, we do have a few laws that could justify the police's actions. First the 13th amendment bars slavery in any form within the United States. Thus the Demon cannot own the possessed individual. The demon must therefore enter into contract where the possession of the individual is appropriately compensated. Consider it similar to a landlord and a renter, where a contract is signed by the two acknowledging a legal agreement including duties and responsibilities of both individuals as well as the compensation of the possessor to the possessee. The demon's claim can be assessed if he is able to temporarily relinquish his hold on the individual so that the police and priests can properly assert that the individual did enter into the contract in good faith.
Now, the tricky part is assessing that the contract was a fair bargain. While I would hate to be accused of being an anti-demonite, demons do have a reputation of entering into deceitful contracting laws. If the individual did enter into a contract with the demon, however, did not comprehend that possession was part of his end of the negotiation, it could be held that the demon entered into a deceptive or fraudulent contract with the individual. Contract Law in the United States typically goes in favor of the defendant (the individual in our scenario) if the language was deceptive or predatory in nature and the defendant was not aware of the negotiate bargain was against his interests.
As the Demon is claiming rights under the United States, I would hardly think a rational modern day court would oppose such a request provided the creature was capable of understanding the rights afforded by the constitution. However, as such, any agreement made in the jurisdiction of a US court would mean that the contract is reviewable under the U.S. Legal system as opposed to the... Damned Legal System (The Court of Hell, lest anyone find room for an Evil Lawyer Joke). As such, the contract can be ruled null and void in part or in full by a Judicial Lawyer.
I would hold that while the Demon does have rights, the contract can only be enforced if the possessed entered into the contract of his own free volition and fully aware as to the terms he was agreeing too.
Should all this be in order, I would like to point that the demon was discovered in the process of booking for an arrest for a possible criminal action. As we would not hold the landlord accountable for a criminal action committed by a tenant on the landlord's property, if we presume the landlord was not aware of such action. Since the demon is under arrest, the police would have the right to remove the demon from his place of residence (the possessed) and hold him in proper incarceration facilities until such time as the demon can appear before a judge to enter his plea and have appropriate bail set. The demon can be asked to come quietly or be forcibly removed from said residence by means of force, both natural and supernatural, if he refuses to willingly surrender to peace officers. Furthermore, the assumption of a fair contract in place, the demon may be held liable for damages to the property incidental to his arrest insofar as described in the contract.
As to the nature of the injuries, the demon does not appear to have received lasting permanent injury. As part of booking procedures, criminals are subject to a number of uncomfortable procedures to acquire possible evidence to identify the nature of the arrested. The demon has not presented any evidence that lasting physical harm was caused by these tests to ascertain his nature nor damage to the possessed. The court would probably find that any utterances of confessions, threats of apocalyptic doom, deciphered speaking of tongues, or general unpleasant language found only in the darkest depths of hell or Enlisted Navy Personnel's minds shall be inadmissible as evidence as they could be construed as compelled testimony. As established, the demon does have full rights under the United States Constitution including his Fifth Amendment protection against self-incrimination.
Edit for Addendum:
In consideration to @CreedArcon's concern, the first amendment holds that congress shall make no law establishing a religion nor infringing upon the free exercise of religion. The finding of the mock-court is the police in this scenario did limit to solely Judeo-Christian demons in testing for possible demon. The defendant reacted to confirm at least one test of an indeterminate religion that has yet to be disclosed to the mock court. The defendants' first amendment rights thus were not violated in this respect. The employ of tactical exorcism units is also acceptable as the government has allowed chaplain services to conduct religious services for troops and police units.
The mock-court also finds against Mr. Arcon's assertion that the defendant must provide papers stating his intent. As with all criminal cases the defendant is presumed innocent until proven guilty. Additionally, the defendant has a fourth amendment right to be secure in his papers and possessions. Thus the mock-court finds the defendant need not provide evidence. Assuming that the defendant was properly read his Miranda rights at arrest, and admitted to the police that he had a contract drawn up to his possession of his host, the mock-court finds sufficient evidence to issue a warrant for said document. Additionally the mock-court finds sufficient evidence to issue warrants to search for evidence related to the crime which the defendant was arrested for. The accusation of Violent over throw of the United States Government (by means of raising Hell) requires evidence that proves intent beyond reasonable doubt and the burden of this proof must be met by the prosecution. While Mr. Arcon believes that Hell may be aiding and abetting a crime and thus has liability the Mock-Court finds that if we do not afford the defendant his right to proper defense, there will certainly be Hell to pay.