There are well known objections to type-type identity (IEP) such as multiple realization and Kripke's argument, but I don't know any objections acceptable (by the majority of philosophers) to this identity version of physicalism. Are there any serious problems in identifying a particular mental event with some physical event?
Identity Theory is a claim that the brain state is identical to the conscious state: https://iep.utm.edu/identity/ Philosophers have treated Identity Theory AS a theory, and tried to falsify it. There are a variety of problems with it that have been identified:
Multiple Realizability. We experience a thought over time, say 2+2=4. Over that time, we have synaptic connections dissolve, and new ones forming, repeatedly. We also have continually varying patterns of energy activation passing thru our brains. Neither the network, nor the activation pattern, remain constant, despite the constant thought. This though therefore CANNOT be identical to either a brain network structure, nor an activation pattern. We are also able to communicate this thought to others, despite their different neural structure, and activation patterns.
Identity of Indiscernibles: Qualia are not indiscernible from brain states. Identity Theory fails the basic criteria of identity.
Physicalism has adopted several different strategies to address these failed falsification tests. The most common is Functionalism. Functionalism ALSO is an Identity Theory, but what is claimed to be identical to consciousness is a function. In my example, the function of 2+2=4 is asserted to be identical to the awareness of 2+2=4.
In Functionalism, functions ALSO have an identity to brain states, but that identity is the token-token identity your question references. Many brain states are postulated that can produce slightly different 2+2=4 functions. HOWEVER, the brain is postulated to act BASED on functions. Somehow, the brain is aware of the large suite of tokens that can produce the related 2+2=4 functions, and keeps its status within that suite. So functionalism postulates that the Functional family resemblance is what is causally effective in bounding the states the brain enters. Token identity itself is casually irrelevant, and cannot EXPLAIN anything. The family of functions are asserted to the type-type identical to consciousness, and therefore this type-identity is the explanation of consciousness.
Objections to functionalism:
The primary objection is the other side of the coin to multiple realizability, it is a surplus of functions relative to consciousness. One can write 2+2=4, design an and gate structure that executes 2+2=4, write a digital code that executes 2+2=4, design a neural net that generates 2+2=4, etc. Yet, 99.999% of these functions will not be conscious. Functionalism is not a sufficient condition for consciousness. INCLUDING IN OUR BRAINS! We process 2+2=4 in many cases UNCONSCIOUSLY -- hence even the family of related 2+2=4 CANNOT BE IDENTICAL TO CONSCIOUSNESS.
Our brains don't actually DO functions! We humans developed the concept of functions from introspection on how we do analysis. We then created digital computers to do this sort of stepwise analytic processing. BUT -- when we examine our neurology, we are wired COMPLETELY DIFFERENTLY than our digital computers, even though our digital computers were structured on how we think we think. Our brains are wired with as associative neural nets, which are analog, not digital. Functions are not discernable at all in our wiring. Daniel Dennett postulates that we run a "virtual" digital computer on our analog neural net structure. Interestingly, much recent work in Artificial Intelligence involves running virtual neural net processing on the actual digital wiring structure of digital computers. BUT back to Functionalism -- if we don't have physical functions in our neurology, yet we are conscious, then functions cannot be a physical explanation of consciousness.
These objections have generated a variety of proposals to get around them.
Paul Churchland proposed that the operation of a recursive neural net IS consciousness -- in an effort to revive neural identity theory. This proposal falls afoul of the first objection to functionalism -- our brains do recursive neural net processing all the time, for basically everything, yet consciousness is a rarity.
Higher Order Processing, and Global Workspace Theory, and Strange Loops, were all proposals to limit the TYPE of functions that could be identical to consciousness. the problem for each is that we have instantaneous conscious qualia that is not higher order, global, etc., AND we have higher order processing, globally shared info, etc. that are not conscious. These narrower functions do not match the experiences of consciousness in functional objection 1, AND do not address functional objection 2 at all.
Integrated Information theory is a proposal that Functional Identify requires, additionally, that the substrate the function is run on must have a high wiring uniqueness (high Phi) in order to generate consciousness (or that the consciousness generated is proportional to Phi). IIT runs afoul of both objections to functionalism -- we don't run functions, and most of our processing is completely unconscious despite being run on either high or moderate Phi wiring.
These various failures to make Identity Theories work, are why most physicalists today DON'T hold by an Identity Theory, and instead are emergentists. Emergent physicalism holds that some neural structures, or possibly some types of neural processing, lead to consciousness emerging. In emergent physicalism consciousness is not IDENTICAL, but a CONSEQUENCE of brains.
I hope this discussion of Identity Theory has made clear the problem with token Identity. Theories of consciousness seek to explain why we are conscious, and for physicalist theories, how the brain maintains our being conscious. Token Identity removes any predictive power from Identity theory, making it an untestable claim, with no predictive or explanatory utility. If Token Identity is all one has, there is no way for a brain to maintain consciousness or functional stability, and neither our functional behavior, nor our conscious awareness are explainable. And in particular, their stability and apparent effectiveness over time is counter-predicted.
In Popperian philosophy of science terms, the token-token claim is an effort to maintain a dogmatic assertion of Mind/Brain identity, while vitiating it of any testable consequences, due to every actual predictive consequence of an actual Identity Theory being refuted.
Your question, as I read it, seems straight forward:
Are there any serious problems in identifying a particular mental event with some physical event?
I think there plenty of philosophical objections to identifying mental events with physical events IF one ignores naturalistic epistemology. That being said, if one embraces it, then one clearly has to accept neuroimaging as the nail in the coffin of any philosophical theory that attempts to refute the general equivalence between mental and physical events. Neural correlates of consciousness are simply too irrefutable to deny statements like "physical events in the brain are strongly correlated to mental events".
I think some philosophers have an axe to grind, as Conifold noted, with some operating principles of physicalism, particlarly those who want to validate some non-empirical transcendental reality. In this case, if one were to accept that NCCs fundamentally show an obdurate metaphysical relationship (of any kind), it becomes much more difficult to allow metaphysical presupposition that posit additional realities. For instance, the need for a soul is obviated by NCCs since they're both non-empirical (to orthodox formulations of empiricism that reject paranormality) and non-parsimonious.
Beyond the powerful correlation, however, all is fair in love, war, and metaphysics. In no way do NCCs endorse particular views in regards to shed light on mental causation, or weigh in on the spectrum of positions from subjective idealism to eliminative materialism, other than to suggest that one can claim that there are no known instances of mental events without the appropriate physical events that we have come to know in love that underlie thinking. That is, clearly the affecting the brain affects the functioning of mental events. And then that fact is free to be used strategically.