Am I committing a sin by refusing to have a third child, despite my wife's desire for one, due to the difficulties I face in raising our two current children and my disagreement with her about the method of upbringing?

A husband does not have the right to prevent his wife from conceiving, as it is a shared right between the spouses, and neither of them may prevent the other unless there is a valid excuse. The reason mentioned in the question is not considered a valid excuse. Therefore, the spouses must reach an understanding and cooperate in raising children, seeking assistance from Allah Almighty.