Can't shake doubts about quality of research, but want to continue
If I have understood your question correctly, you are doubting that the methods used in Paper A (and followed in Paper B) are "correct." The question of whether or not Paper A's methods are "good enough" to get the "right" answer to this problem is an empirical one you can investigate. 

You should consider proposing and writing a paper where you look at a similar problem as Paper A with two methodologies and see if there are different results. You could do what your advisor wants and also solve the problem a different way and compare. This would be a valuable contribution and would make your point for you (or not).

You could do this while also starting Paper C to make your advisor happy. You may also consider publishing Paper C with a simple method as a "first pass" and including caveats about the method and how it may be affecting the results. Then when you get the results to the comparison paper it will become obvious if Paper C also needs to use the new method, and you could publish a follow-up paper (or add to Paper C if it is not yet submitted).