Which is more proper "rarest" or "most rare"?

There's no such thing as "more proper" (not that it should've been *properer !) The "correct" superlative of rare is rarest as we know. Use it by default.

However, more/ most + (positive) also exist & have their uses even in cases where the word already has regular comparative & superlative.

Care needs to be taken so the most is not misunderstood to apply to the noun, though.

In the instant case, it is rarest, as the other form is neither needed nor approriate.

If the word has one or two syllables, comparative form is formed by suffixing -er and superlative form is formed by suffixing -est. For words with three or more syllables, it's usually more xyz or most xyz. When we ask ourselves the superlative of rare, rarest comes more naturally. But for most, its properer to use more proper and most proper.

I think there's a reason why words with less syllables are suffixed with more/most to create their comparative or superlative form. E.g. when we say : "This chimp species is more rare than that.", we are actually adding another word "most" to convey the message. Using -er and -est shortens it.

EDIT

You can also look at the historical popularity of "rarest" and "most rare" through Google n-gram veiwer: link

The terms most rare, rarest, more rare, and rarer have all been in use for centuries. Here is the Ngram chart for the years 1800–2005 (search result matches go back to the 1500s):

The chart suggests that rarest (red line) and rarer (yellow line) have been more popular (not popularer) forms than most rare (blue line) and more rare (green line) for at least 150 years. Nevertheless, given their consistent occurrence in published works over so many years, I wouldn't argue that any of the four forms is wrong.

Perhaps one reason for the persistence of the more rare and most rare forms is that the contrary forms are less rare and least rare. Also, to the extent that comparatives of rare arise in contradistinction to comparatives of common, more rare and most rare may benefit from the relative popularity of more common (green line) and most common (blue line) as against commoner (yellow line) and commonest (red line):

Here, commoner is the least common of the four terms despite (or perhaps partly because of) the fact that it has an identical noun form.

Multiple people have mentioned the accurate, and pedantic, rule that single-syllable words use a suffix not "most". They are correct. Nevertheless, I find the "rule" annoying and unconvincing.

To me, the most powerful reason to use "rarest" rather than "most rare" is Rule 17 of The Elements of Style by Strunk and White: Omit needless words. I personally think of this rule with a phrase in the description of the rule: make every word tell.

Let us pretend that the Ngram data were reversed: I would still advise you to strongly prefer "rarest" and to only deviate when the word interferes with the poetry of the sentence.

Make every word tell.

Ah hah, I like this.

"I would still advise you to strongly prefer "rarest" and to only deviate when the word interferes with the poetry of the sentence."

I am an analytical, logic, and math based thinker and yet I felt that most of these answers preferred to use logic over the real point of language. It is not only to communicate but to also engage and dare I say spark the neurons in the artful, creative portions of the brain. In fact the goal of communicating, and the goal of engaging are not contradictory goals. By engaging the reader the meaning of a piece of communication is conveyed even more accurately and precise communication is enhanced. As long as "most rare" is not confusing, and if the writer feels it adds poetry to the sentence then I think they do an injustice to the language, and to their creative soul, if they use "rarest" because a text book says it is the proper use.

In addition I would add that the usage over time is interesting but not informative in providing an answer to the original question. If so this becomes a self-fulfilling prophecies and drives us toward a more uniform but less interesting language. Save this sort of analysis for German, not the language of Shakespeare. Those are my two cents, have a wonderful day.