Do we use another "the" if a proper noun already begins with one?

The does work that only needs to be done once, as necessary, for each occurrence of a noun in a sentence:

definite article

1. (used, especially before a noun, with a specifying or particularizing effect, as opposed to the indefinite or generalizing force of the indefinite article a or an):

the book you gave me; Come into the house.

2. (used to mark a proper noun, natural phenomenon, ship, building, time, point of the compass, branch of endeavor, or field of study as something well-known or unique):

the sun; the Alps; theQueen Elizabeth; the past; the West.

3. (used with or as part of a title):

the Duke of Wellington; the Reverend John Smith.

4. (used to mark a noun as indicating the best-known, most approved, most important, most satisfying, etc.):

the skiing center of the U.S.; If you're going to work hard, now is the time.

5. (used to mark a noun as being used generically):

The dog is a quadruped.

6. (used in place of a possessive pronoun, to note a part of the body or a personal belonging):

He won't be able to play football until the leg mends.

7. (used before adjectives that are used substantively, to note an individual, a class or number of individuals, or an abstract idea):

to visit the sick; from the sublime to the ridiculous.

If the name of an entity contains The, that entity has been identified definitely in its name, and the work of the is completed. No repetition is necessary.

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dictionary.reference.com

I'm inclined to say drop the first "the" as well, but as it's common practice to italicise titles of works, you could get away with "the The Republic DLC" (consider "the The Count of Monte Cristo manuscript"). I would still prefer to reword the sentence rather than use this approach, e.g. "...if the player owns the DLC 'The Republic'" or "...owns the DLC The Republic".

'The The' is never correct except as the name of a band.