Just correct him. What is wrong with argument? Is he going to physically attack you?

"you catch more flies with honey than vinegar."

The fun part about that truism is that when actually tried. You find out the flies love vinager, Far more than honey, something about the smell?

It could be said the other way around, which is also true, the best way to get someone to respond is not a well structured question but by being wrong.

Praise in public, critique in private.

I wouldn't say it's about compromise, or even avoiding conflict as such. The first thing we must always remember is what is true, right, and just, for our own sake above all. One ought never to compromise morally - you have lost the bigger game by doing so - and conflicts are simply a matter of fact, so it's all a matter of avoiding pointless engagement in conflicts that don't serve the good.

Now, unreasonable and malicious people will transgress the moral, so the question is then: if I must deal with this person, how do I do so as fruitfully as possible?

What egotistical people don't realize is that their vanity is not a strength, but their weakest asset. Humility is a kind of invulnerability, so to speak, because all humility is is a disposition to recognize the truth and to conform to it. Egotistical leaders are highly motivated by a need for validation, for reputation, and for "glory". So, if your boss is egotistical, you should manage it and leverage that ego for the intended aim. Don't take things personally. Focus on the rational goal you are all aiming at. Plant seeds in conversations so that the leader can permit himself to think your ideas are actually his. Surrender the spotlight to him. Let him bask in the blinding limelight. Attention is a prison; if he wants it so badly, let him have it. Magnanimous men are above such things.

Never resort to trickery. This will burn you in the end, especially when the truth comes out. Never flatter or resort to adulation or sycophancy - it is smarmy, demeaning, and undignified; you see plenty of this in Trump's circle - but recognize actual strengths of the leader.

Maintain composure (do not react with fear or eagerness to please), distance, and politeness. Be patient: this allows the egotistical leader's passions to cool down, creating more space for the rational. Frame acknowledgement and concession to someone else's good ideas and advice not as a loss - which is how egotists see things - but as a magnanimous act on his part. Magnanimity is foreign to the egotist, but his love of reputation and a desire to be seen as magnanimous suffices for practical purposes.