I see that a proposed term in one paper is State Noncooperation. I'm not sure how widespread that is. I'm also having doubts as to its generality, because it seems to mainly apply to the case where the state has actually passed laws explicitly limiting some forms of cooperation. And this is actually not as recent as one might think with respect to immigration; a 2008 paper noted:
The cooperation debate in the immigration context has immediate, practical significance. Currently, some 49 cities and states have laws that limit the authority of their police and other employees to cooperate with federal immigration law enforcement. These local non-cooperation laws prohibit local government employees from reporting to the federal government undocumented persons they encounter in the course of providing police protection, health care, education, and other essential local government services. At the same time, the federal government has passed laws requiring the very same cooperation that local governments have prohibited: under federal law, local governments must allow their employees who want to cooperate with federal immigration enforcement on a voluntary basis to do so. This federal push to get local governments involved in immigration law enforcement has intensified after the attacks of 9/11.
Under the current legal framework, the federal government wins. Its exercise of its preemption power faces no Tenth Amendment challenge because, unlike Printz and New York, federal cooperation laws do not commandeer local governments into passing legislation or enforcing federal schemes. [...]
In April 2002, the Justice Department wrote
(but did not release) a legal opinion stating that cities and states have “inherent authority”
as sovereigns to enforce immigration laws. Then-Attorney General John Ashcroft
followed up with an invitation to local police to enforce immigration laws as part of “our
narrow anti-terrorism mission.” And members of Congress have drafted legislation to
give financial incentives to cities and states to enforce immigration laws (and financial
penalties for those which refuse). The goal: to dramatically multiply the enforcement
power of federal immigration authorities by enlisting the aid of local police and other
local authorities, who are already “on the beat” in America’s cities and towns.
While some local governments enthusiastically embraced the opportunity to
enforce immigration laws, others refused to become involved, passing laws that limit, to
different degrees, their authority to cooperate in immigration law enforcement (“non-cooperation laws”).
The language and scope of these non-cooperation laws vary. A
typical non-cooperation law was that passed by the state of Alaska in May 2003,
prohibiting Alaskan agencies from using state resources to enforce immigration laws. In
Fresno, California, the non-cooperation law is much more specific: prohibiting the police
from reporting undocumented immigrants to federal immigration authorities in cases
where no other crimes have been committed. And Seattle’s ordinance, passed in January
2003, cuts off local cooperation at an earlier pass by prohibiting police officers and other
city employees from even inquiring about the immigration status of any person, unless
otherwise required by law.
But at least the terminology "non-cooperation laws" appears somewhat established when such laws do exist. A Senate report from 1995 also makes use of this term (non-cooperation laws), but more often mentions "non-cooperation policies". The latter is obviously broader in scope.
As an aside: while not obliged, states are allowed to cooperate in federal law enforcement although the means seem to need be specified in each statue:
In fact, there are two types of public enforcement. Many federal
statutes authorize civil enforcement by both a federal agency and the
states, typically through states' attorneys general. State enforcement
provisions appear most frequently in federal laws designed to protect
consumers, such as the recent Dodd-Frank financial overhaul bill.
Proponents of state enforcement emphasize its potential to buttress
federal efforts by putting more "cops on the beat." [footnote:] statement of Sen. Mark Pryor arguing that Consumer Product Safety Improvements Act (CPSIA) "ensures that [state attorneys general] can act as real cops on the beat, looking out for consumers and restoring confidence in the marketplace".
Also note that the legality of state non-cooperation laws (as opposed to mere ad hoc non-cooperation) is currently making its way through the Federal Circuit, at least in the case of California.