Excellent question. The article "the" makes the gerund "managing" more clear as a part of the dependent clause that begins with "for" in the sentence. Hence, the most natural phrasing of the options you listed was probably the penultimate one ("We need an experienced employee to manage our portfolio."), since it removes the clunky prepositional phrasing that is creating potential confusion with the article. The final one ("We need an experienced employee for the management of our portfolio"), in which the word "management" replaces the potentially-confusing gerund "managing," also works. Please let me know if you have further questions, or if this answer is unhelpful for any reason.
Best,
Tori