An epitome—grammatically correct?

Grammatically there is nothing wrong with saying "an epitome". However there are logical issues with it.

"epitome" is defined as

A person or thing that is a perfect example of a particular quality or type.

Most people would consider that only one thing can be a perfect example of anything, since unless two things are identical one must be a better example, and the other is not perfect.

You could theoretically say that one of several absolutely identical things is "an epitome" of something.

The Ngram chart for the frequency of "an epitome of" (blue line) versus the frequency of occurrence of "the epitome of" (blue line) in published works included in the Google Books database shows a striking reversal of fortune over the 120 years from 1880 to 2000:

The shift in relative popularity of the two phrases may not be as simple as it looks in the Ngram chart, however. One crucial complication is that epitome has three meanings, and one of those meanings is very different from the other two. Here is the entry for epitome in Merriam-Webster's Eleventh Collegiate Dictionary (2003):

epitome n (1520) 1 a : a summary of a written work b : a brief presentation or statement of something 2 : a typical or ideal example : EMBODIMENT {the British monarchy is the epitome of tradition —Richard Joseph} 3 : brief or miniature form — usu. used with in

However inclined one might be to use the, rather than a, with epitome in sense 2 of the word, no such inclination attaches to senses 1a and 1b. (Sense 3 falls outside my inquiry because, according to the Eleventh Collegiate, it tends to take the form "the/a epitome in." If the dominant sense in which epitome was used in the period 1750–1920 was "a summary of a written work" or "a brief presentation of something," but the dominant sense of the word after 1940 was "embodiment," it would not be surprising to see the lines for "an epitome of" and "the epitome of" cross at about the year 1940, as they do in the Ngram chart.

In this regard, it seems worthwhile to note the strong preference in English for "the embodiment of" (red line) over "an embodiment of" (blue line), from about 1840 onward:

When did the "embodiment" sense of epitome emerge in English? The shifting and expanding definitions in Merriam-Webster's entries for epitome offer some insight into this question. From Webster's Collegiate Dictionary, first edition (1898):

Epitome, n. A brief summary ; an abridgement ; a condensation ; synopsis.

The entry for epitome in the second edition of Webster's Collegiate (1910) is unchanged from the one in the first edition. From Webster's Collegiate Dictionary, third edition (1916):

epitome, n. 1. A brief statement of the contents of a topic or a work; an abstract. 2. A compact representation of anything.

The entry for epitome in the fourth edition of Webster's Collegiate (1949) is unchanged from the one in the third edition. From Webster's Collegiate Dictionary, fifth edition (1936):

epitome n. 1. A brief statement of the contents of a topic or a work; abstract. 2. A part which represents typically a whole.

The entry for epitome in the sixth edition of Webster's Collegiate (1949) is unchanged from the the one in the fifth edition. From Webster's Seventh New Collegiate Dictionary (1963):

epitome n 1 a : a summary of a written work b : a brief presentation or statement of something 2 : a typical representation or ideal expression : EMBODIMENT 3 : brief or miniature form

Evidently, at some point between 1949 and 1963, popular use of epitome in the sense of "embodiment" became so substantial that Merriam-Webster decided to add that meaning in its dictionary entry for the term. Obviously this did not mean that the earlier meanings vanished from usage, but it does mark the dictionary's recognition of the rising popularity of the relatively new meaning.

Since the Ngram chart shows the frequency of "an epitome of" dropping rather sharply for at least four decades before the frequency of "the epitome of" begins to rise appreciably, it seems quite possible that epitome in its "abstract or synopsis" sense was fading in popularity independently of the emergence of the "embodiment" sense of epitome. On the other hand, the diminishing use of epitome in its earlier senses may have provided an opening for the new meaning to assert itself with less ambiguity—and of course, speakers' and writers' tendency to attach "the" rather than "a" to epitome when using it in this newer sense would have helped reinforce its distinctness from the earlier senses of the word.

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The OP asks whether one can properly say "an epitome of" when the sense of epitome is "embodiment," or whether the only proper phrasing in such situations is "the epitome of." A look at some recent published works indicates that both the indefinite article and the definite article are legitimate options. Consider, for instance, Ansbert Ngurumo, John Magufuli: An Epitome of Cowardice (2019) or G. Venkataramana Reddy, Alayam the Hindu Temple: An Epitome of Hindu Culture (2016) or this excerpt from I.M. Ezergallis, Male and Female: An Approach to Thomas Mann’s Dialectic (2012):

They [a twin brother and twin sister] are aware of being a pair from birth. Identical, differing only in Wiligis' being an epitome of physical manhood and possessing the psychological attributes that go with it, and Sibylla being a comparably perfect incarnation of femininity, they are constantly together as children and adolescents.

Or this excerpt from Dorothea Tanning, Between Lives: An Artist and Her World (2011):

In August 1949 we, too, are there with him. An epitome of desuetude, the café is deserted except for the surrealists. It is a dim dark-wood-and-mirror kind of place indistinguishable from thousands of other cafés across the city. There sit the surrealists, votaries all, typecast around the table.

In all of these instances, epitome means "embodiment," not "summary" or "synopsis" or "brief statement." It thus appears that something can legitimately be "an epitome of" something else or "the epitome of" something else, depending on the speaker's or author's intention and preference.