Idiomatic expression for a difficult choice

Dilemma, "a circumstance in which a choice must be made between two or more alternatives that seem equally undesirable" is frequently used for situations somewhat like you describe. (Note, wiktionary also says "the sense of a difficult circumstance or problem is considered non-standard" by some.) Etymonline says dilemma arose in the 1520's,

from L.L. dilemma, from Gk. dilemma "double proposition," a technical term in rhetoric, from di- "two" + lemma "premise, anything received or taken," from root of lambanein "to take" (see analemma). It should be used only of situations where someone is forced to choose between two alternatives, both unfavorable to him

which supports the notion that the choices are unfavorable. Unfortunately, this word does not have the connotation of the choices being mostly favorable and partly (but seriously) unfavorable, which your question calls for. Perhaps you could use delicious dilemma or tasty dilemma.

I guess the word/phrase you are looking for is "toss-up", as quoted from http://www.learn-english-today.com:

When there are two options or possibilities to chose from, and both are equally good, the choice between the two is referred to as a toss-up (like tossing a coin). "Both boxers are in excellent condition. It's a toss-up which of them will win the match."

You might consider 'It's six of one, half a dozen the other.'

You are "between a rock and a hard place".

You might be caught between the Scylla and Charybdis:

Being between Scylla and Charybdis is an idiom deriving from Greek mythology. Several other idioms, such as "on the horns of a dilemma", "between the devil and the deep blue sea", and "between a rock and a hard place" express the same meaning of "having to choose between two evils".

I think OP has really gone beyond any standard idiomatic expressions by specifying two different largely unrelated constraints.

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• both choices "are desirable", but there's the implication that you can only pick one.

The standard expression there is you want to have your cake and eat it. Often poorly understood, this means you can't continue to own your cake if you eat it, because then it's "gone".

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• Both choices "have a tiny part which is most undesirable".

Here, each choice has pros and cons. The fact that the "pros" and the "cons" may be very unequal is irrelevant to the use of this expression.

Almost made for it is "on the horns of a dilemma." The archetype of this phrase is from the Old Testament of the Christian faith. Specifically, the story of Joab in First Kings, Chapter 2, verses 1-46. I won't give the story away. But Joab was faced with a terrible choice. And to try to delay it, he fled into the temple and grabbed hold of the horns of the altar (pictured). And there he refused to leave.

By extension, somebody who has a difficult decision and clings to the decision trying to delay it, is "on the horns."