An objective, "mind-independent", view is a staple of physics, but so is the possibility of observation. The information lost in burning is not entirely unrecoverable, it is recoverable in theory, and some futuristic nanotech may be able to recover it, with a lot of effort, even in practice. If we do not go to extremes then it is well-known that information that was not scrambled radically enough can be recovered by crafty experts even today. Breaking the Humpty-Dumpty increases entropy, and hence "loses" some information, but Humpty-Dumpty can be put together again. This is not much different from acknowledging reality of the past, etc., we may never know what Aristotle did on his 20th birthday, but realists postulate that there is a fact of the matter as to what that was. As Dupre puts it somewhat sarcastically in Metaphysical Disorder and
"...some set of facts that could be known that would permit the inference of the macroscopic from a sufficient knowledge of the microscopic. Perhaps we could not, even in principle, know these facts. But God, I suppose, would need merely to exist in order to know them."
But sarcasm aside, taking the "God's eye" view is typical among mainstream realists when interpreting claims that would not be sensible otherwise. E.g. in Relativity of Simultaneity and Eternalism Peterson and Silberstein invoke "Newton's god" (NG) to distinguish eternalism and presentism about time:
"There may be some who believe that NG is not a suitable tool for dealing with the presentist/eternalist distinction; in particular, one might findnd our NG question-begging since a god's eye point of view might somehow allegedly violate basic tenets of SR, however, one must note that by hypothesis NG is removed from the 4D-manifold (space-time) that she observes.... It would be absurd to argue, therefore, that two perspectives as different as these are, are in fact, metaphysically and empirically equivalent in principle; such a claim could only be sensible if one assumes a spatiotemporal-anthropocentric verificationism and there is no non-question begging reason to do so."
The second law of thermodynamics is an emergent statistical law that applies to macroscopic systems with large number of microscopic objects. It can be violated, and was observed to be so in some experiments, but the probability of that decreases with the number of objects involved. It does not apply to systems (taken separately) with influx of energy that prevents them from reaching equilibrium, e.g. to the gravitational accretion of matter that formed the planets of the Solar system. In short, there is plenty of room to validate the second law for all practical purposes, while maintaining whatever one pleases at the fundamental level. There is also a difference between physicist's and colloquial meaning of "information", a string of symbols gibberish to a human knower, is full of "information" in the sense of being correlated with the part of the environment that "created" it, but I do not think that this difference is relevant here.
Determinism is a bigger problem. It is a mathematical consequence of quantum mechanical formalism that unitary evolution by itself conserves information (more precisely, the von Neumann entropy). It is the non-unitary "collapse of the wave function" that creates problems, which is perhaps why most quantum information theorists, like Deutsch, subscribe to the Everett interpretations of quantum mechanics. There there is no collapse, and decohered states, which create an impression of it, can in principle recohere, unlikely though that is. Everett's is a strange "determinism" though, his branches realize every possible outcome rather than predetermine one, hence each branch "creates" itself through its past "choices". Only the global state is predetermined, but it is everything whatever anyway. Essentially, the unobservable branches are postulated so that there is somewhere for the "lost" information to go (or come from).
In statistical interpretations "collapse" amounts to observer specific reconditioning of probabilities, and so any information "loss" it entails will also be observer specific, it does not happen "objectively". Under objective collapse, and indeterministic interpretations of classical physics, one would indeed have to reject the conservation of information. If a ball rolls to the top of the Norton dome and then stops there, there is no recovering afterwards from whence it came and how long it stayed on top. This means of course that quantum mechanical formalism per se is neutral on the issue.
P.S. Time reversibility by itself does not entail determinism, and the Norton dome is a counterexample. Time reversed solutions to the ones mentioned above, when a ball sits on top of the dome for arbitrary time, and then rolls off in arbitrary direction, are also solutions (the shape of the dome makes equations of motion non-Lipschitz, and uniqueness of solutions is lost). What it does entail is that if information can be destroyed it can also be created "out of nothing", like the ball's elapsed time and direction of movement. The peculiarity of the dome is that self-initiation of motion requires no energy input here, not even an infinitesimal push. This is close to what free will libertarians want, free will acts as causa sui, and hence injects new information into causal chains. I should mention that many physicists consider the Nortom dome "unphysical", and "free will creates information" is likely to be as controversial as "consciousness causes collapse".