What does Mr. Trump’s “inner rabbit” mean?

I read Ms Collins every Thursday and Saturday. She is one of my favorite writers. Despite my fondness for her, I will endeavour to have my answer devoid of praise or criticism of her talents, unlike Mr. Blow's very biased response.

If you read the full article, it would appear that she is referring to her belief that he has no "political courage":

The great theme of the night was things that Donald Trump said that he now doesn’t remember, or didn’t necessarily mean. This happens all the time. Either our great business genius is incapable of mental fact-checking, or he has about as much political courage as a rabbit.

So in this case, Collins apparently feels that Trump is showing a lack of courage by toning down his comments on the Pope. Initially Trump said of the Pope "For a religious leader to question a person’s faith is disgraceful." But later he had kinder words, as quoted by Collins.

It may also help to be aware that Ms Collins has a history with Trump:

During one down period, I referred to him in print as a “financially embattled thousandaire” and he sent me a copy of the column with my picture circled and “The Face of a Dog!” written over it.

The notion of an "inner X," where X is represented or treated as a complete being hidden within a person who on the outside may appear quite different from this inner entity, goes back many decades. One version of the "inner X" trope that emerged in the early 1900s is "inner warrior," a term used repeatedly in theosophical literature. For example, from Jerome Anderson, "The Battle with Self," in Universal Brotherhood Path (December 1900) [snippet view]:

It must be recognized, also, that the brain-mind, which thinks our ordinary thoughts, is the servant and representative of the lower self, and must be held in strict subjection. Ninety-nine out of every hundred of our thoughts are the direct result of the things reaching us by means of the senses. The brain-mind may be made the servant of the Higher Self, but this can only be accomplished by being constantly on guard under the guidance of the inner Warrior.

(You can read Anderson's entire article elsewhere online.)

By far the most common "inner X" in recent years is "inner child." This notion began (in the nineteenth century) as a way of distinguishing between the outer person of a child and the inner thoughts and feelings of that same child. For example, from Charles Williams, George Mogridge: His Life, Character, and Writings (1856):

Agile as was that little frame, far greater still was the activity of the inner child. Though all the mental powers were in exercise, the imagination was always strongly disposed to assert its supremacy, and often was it successful.

But in more recent times, popular psychology has made an industry of analyzing the needs longings, and protests of the "inner child" of adult human beings. Discussions of an "inner child" that exists within an adult began to appear in the early 1950s (if not sooner). From Rhoda Kellogg, Babies Need Fathers, Too (1953):

Becoming conscious of this inner child and frankly admitting its existence helps us to deal with it more rationally. Denying that we tend to behave childishly under adequate pressure is only to strengthen the power of that immature little creature inside us who causes most of our troubles. A father must try to figure out how the child within him still reacts to his parents, to traditions of family life, to the idea of authority in general, and to the idea of his own parenthood.

In the context of "inner X" as a psychological trope, "inner rabbit" can mean a number of things. JeanMarie Brownson, Dinner at Home: 140 Recipes to Enjoy with Family and Friends (2015) uses the term to refer to a delight in greens and vegetables:

It's easy to eat plants when they're covered in bacon and blue cheese, which defeats the point. Same goes for salads swimming in dressing and packed with deep-fried croutons. That's why I like to make fattoush. This Middle Eastern toasted bread and vegetable salad satisfies my inner rabbit perfectly.

Natalie Haynes, The Furies: A Novel (2014) suggests that it finds expression in frolicsome leaping:

When actors are in training, they're often asked to think animalistically: which animal are you, how can you convey that without words, and so on. Much leaping about the room channelling an inner rabbit ensues. But on that cold, damp morning, when I walked into his Gray's Inn Road office, Charles Brayford made me think that if he had ever decided to search for his inner bunny, he would have eaten it.

But the most common sense of "inner rabbit" is an inner self quivering with trepidation. From Lawrence Rosenblum, See What I'm Saying: The Extraordinary Powers of Our Five Senses (2011):

Just below your awareness, you live in a parallel world: a world where your evolutionary heritage (loosely speaking) keeps you safe. Your inner bat listens to the spaces you occupy. Your inner rabbit listens for threats and anticipates their approach. Your inner dog allows you to determine the location of smells and your inner mouse helps you implicitly use those smells to perceive family, fertility, and reproductive potential. Your inner spider allows you to feel things without directly touching them and your inner firefly helps you sync with people. Your inner monkey helps you recognize intent from faces and effectively mimic the behavior of others. And your inner ferret allows all of these skills to be fine-tuned through the neuroplasticity inherent to nervous systems.

Used pejoratively, "inner rabbit" implies a predisposition toward cowardice that the outer person may conceal. From Robert Gore-Langton, Journey's End: The Classic War Play Explored (2013):

J.R. Ackerley referred [in My Father and Myself (1968)] to his front line fear [during the Great War] as 'the rabbit within.' The moment [Lieutenant R.C.] Sherriff arrived in France [in 1916] his war was a constant battle with his own inner rabbit. It was not just fear for his life, but fear that he would not be able to function properly as an officer. It fed directly into his plot [in the play, Journey's End]. The dread set in very early at Vimy Ridge and he never quite found a way of delousing himself of its effects.

And that seems clearly to be the implication of "inner rabbit" in the editorial column that Yoichi Oishi asks about, given that at one point (as noted in Bumptious Q Bangwhistle's answer) the columnist says, "The great theme of the night [at a town hall meeting broadcast on CNN] was things that Donald Trump said that he now doesn't remember or didn't necessarily mean. This happens all the time. Either our great business genius is incapable of mental fact checking, or he has about as much political courage as a rabbit."

Inner Rabbit is fairly new, and is just about being "timid", rather than loud and banging like he was before.

As: He backed down, chickened, got scared, started wearing brown pants because of the pope, the papal state, and the Swiss guards.

Of course the article writer didn't have enough backbone to call him chicken and resorted to calling him other animal names instead of Chicken.

Yes, I think that it refers to Trump's attitude who, despite his 'arrogant' ways to express his thoughts, in this case he lacked the courage to go ahead with his attack on the Pope (probably for political convenience).

The rabbit is a symbol of cowardice, and the author is suggesting that Mr. Trump has shown his cowardice in this specific case:

Coward:

• Word History: A coward is one who "turns tail." The word comes from Old French couart, coart, "coward," and is related to Italian codardo, "coward." Couart is formed from coe, a northern French dialectal variant of cue, "tail" (from Latin cōda), to which the derogatory suffix -ard was added. This suffix appears in bastard, laggard, and sluggard, to name a few. In heraldry a lion couard, "cowardly lion," was depicted with his tail between his legs. So a coward may be one with his tail hidden between his legs or one who turns tail and runs like a rabbit, with his tail showing.

(AHD)