Your sentence is the better way to say that. The first sentence looks wrong. There is no need of 'in'. It has to be We disagree with the way WHICH they handle the bribery affair IN ... but is not used this way frequently.

"We disagree with the way they handled the bribery affair" is correct. "We disagree with the way IN WHICH they handled the bribery affair" is correct. "We disagree with HOW they handled the bribery affair" is correct. ***** "We disagree with the way WHICH they handle the bribery affair IN" makes no sense and is grammatically INCORRECT! The reason "in" is in this sentence is because "way" (or "manner") needed the preposition "in." e.g. "The bribery was handled *in* a poor way," "The bribery was handled *in* an incorrect manner." Putting it on the end of the sentence disconnects it from "way." So "in which" cannot be split; the necessary prepositions always go directly in front of "which." e.g. "at which," "above which," "in which," "through which," etc.

without in is fine