Fundamentally, if stuck to all the requirement, this cannot be done.
But if you were to do it, the most ironcast method would be by demonstrating that the magician understands how the magic works, and can rationalise and predict what will happen when it's used.
At this point, magic can be proven as real, and the magician in question has essentially founded the science of magic, aka theoretical magic in the process of explaining it.
Science as defined by Wikipedia is:
Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe
The idea of something that is by its nature is unscientific, and exists in the world, is meaningless, and suggests a misunderstanding of what the scientific principle is. It's similar to saying that you have water that is free of chemicals ( despite H2O being a chemical, along with anything else built from atoms ), the statement makes little sense.
As for the proposed solution, this has happened in the past! Although the theories put forward were eventually disproven as it was found that other explanations were more likely. For example, theories of Alchemy lead to the theories of chemistry. Thaumaturgy eventually lent itself to the existing fields of philosophy, and most of its aspects were shown to be less than rigorous. Homeopathy is a more recent attempt that has failed scientific enquiry, yet is recent enough to still encounter strong support.
The main difference here is that in your fictional world, this magician has actual magic, and can back up their statements. By explaining the mechanisms by which magic works, and providing empirical evidence to back up said hypothesis, the magic moves from a scientific hypothesis, to a scientific theory.
Without this, the likely reaction is that the magician is a charlatan and a liar, who tries to fool people into believing they have supernatural powers through trickery. Plenty such people already exist in our world, and if the magician demonstrates sufficient magical abilities, they may be seen as using technology as yet to be revealed
Now the question does state that the magic needs to be explained using current science, but this demonstrates a misunderstanding of science works.
Specifically, it has to have a specific scientific explanation. Most scientists believe that everything fits into science, but this magician wants them to know how his magic fits in.
Nevermind that the statement "scientists believe that everything fits into science" is simply false ( dark energy, dark matter, the internal nature of black holes, where gravity comes from, etc to name a few ), the very act of doing this creates a hypothesis on which the scientific principle can be applied. New science has been created.
- Hypothesis: Expected answer to OP question
- Experiment: Test that explanation works
- Result: Evidence that can be used to construct a new theory
Lets say that your magicians only trick was a vast light show in the sky, and your magician decided to tell the scientists that he used a hidden wide field laser to shift the atoms in the air into a bose einstein state usually only seen in laboratories studying quantum states. This seems sensible at first, but your magician has unwittingly made a huge advance in quantum mechanics, one that will need to be reproducible.
Which is another vital element. For your magician to be believed, whatever they say needs to be explained in a way that other scientists can reproduce. If it cannot be independently reproduced, then your magician will be accused of falsifying evidence and lieing, and you're back to square 1
As a further note, if I make a claim that all people with the name Veronica have a secret ability to shoot beams of fire out of their palms that they are unaware of, I have created a Scientific Hypothesis. The current understanding of Science has changed. Granted, that hypothesis will not be taken seriously, and I highly doubt it will stand up when tested. But if it were tested, and reproduced, the general understanding of Veronica fire beams will be the theory.
All in all, if we strictly adhere to the original questions parameters, the answer is no. It cannot be done without fundamentally changing the scientific method, and demonstrates a fundamental misunderstanding about what science is, and how science works by the asker