Too often people are concerned with their reputation at the expense of intrinsic good--virtue. Virtue and its relation to happiness should be recognized first, even if it means damage to one's reputation. Aristotle and Aquinas confirm, "Happiness is man's true good." But, is a good reputation (admiration by others) the way to true happiness? Conscience is ordinal. And, from where does conscience, consciousness derive? No philosopher or scientist, including the neuro-philosophers have come to anything definitive. The person, truly wise, is not always that person singularly capable of most clearly and quickly grasping the truth of fundamental or experimental reason or logic; but, it is more prudent to be intuitively correct; and--most of all--better to know, indeed, there is a correct solution. Consider, 'A' is not non-'A'. There is truth. And, truth in either philosophy or science is not a matter of synthetic notion but of recognition.
Your question is miss-specified. You simply played with words, good and virtuous are synonyms, and you haven't specified the difference or the actor and actions clearly.
Do you mean: "Should someone who philosophizes (loves wisdom) be concerned with having a good reputation or with being correct?"
And not what you wrote: "Should the successful philosopher (one who 'does' philosophy) be concerned with having a good reputation, not with being virtuous (correct)?."
If so, the answer is self-evident as it is contained within the definition.
If your concern is with success, - Should one who successfully loves wisdom be concerned with being correct or having a good reputation? - The answer remains clear by definition.
If you have a different definition of success you must specify.