No, the form "the prior written X's consent" would be wrong, chiefly because both "the" and the possessive ('s) form act as determiners, and you can't have more than one determiner in a noun phrase. It would be like saying, "Please hand me the my hat", which is more clearly incorrect. If you removed "the" and placed the possessive in its position, you would have a correct alternative phrasing: "without X's prior written consent" Stylistically, I wouldn't change the original sentence if the company name is very long.
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