Is a Muslim's abandoning enjoining good and forbidding evil toward a disbeliever, out of fear that the disbeliever might embarrass him or that he might face an embarrassing retort that could lead to disbelief, considered a form of religious compromise at the expense of religion? And what does "religious compromise at the expense of religion" mean, with examples?

Leaving off the enjoining of good and forbidding of evil is a sin, not disbelief, and not every act of flattery (mudahana) constitutes disbelief. Abandoning the enjoining of good and forbidding of evil for aforementioned reasons is not considered disbelief, even if it is deemed unlawful and sinful when it is obligatory. The mere awe of people is not an excuse for abandoning the enjoining of good, for in the Hadith: "Let none of you belittle himself… by seeing a matter concerning Allah about which he should speak, but he does not speak. So Allah, the Mighty and Majestic, will say to him on the Day of Resurrection: 'What prevented you from speaking about such and such?' He will say: 'Fear of people.' He will say: 'It was Me whom you should have feared more.'" In another narration: "Indeed, Allah will question His servant on the Day of Resurrection until He says: 'What prevented you, when you saw the evil, from denouncing it?' So if Allah inspires a servant with his argument, he will say: 'O Lord, I hoped in You and I feared people.'" These Ahadith imply that the impediment to enjoining good should be mere awe, not fear that would excuse one from enjoining it.