To see the distinction between the directives, I think the key to reconciliation is in the Hebrew word used for "lest," which is "pen" in both verses. There are two words that are translated from Hebrew to one word, "lest," in old English which essentially means
perhaps. Pen means 'perhaps no' (preferably here) and the other Hebrew word also translated as "lest" is "alai," or 'perhaps yes,' but neither of these outcomes, to be like him nor that he be wise in his own eyes, is the preferred outcome so the word 'pen' is used.
Looking at it this way:
Do not answer a fool according to his folly (that) perhaps not you will be like him.
And
Answer a fool according to his folly (that) perhaps not he will be wise in his own eyes.
So, it seems that Solomon's observation is, whether you answer a fool according to his folly or we do not answer a fool according to his folly....
1) you will be like him also if you do not answer a fool according to his folly.
and'
2) he will be wise in his own eyes if you do answer a fool according to his folly.
Assuming these are opposite outcomes, let's put both outcomes in their perspective results.
Answer a fool...and: 1) you will be a fool like him and 2) he will not be wise in his own eyes.
Do not answer a fool...and: 1) you will not be a fool like him and 2) he will be wise in his own eyes.
It seems that it's best just to avoid fools.