I am about to submit a manuscript for publication that is useful for solving a certain class of problems in my field. The technique is clearly applicable to a certain longstanding problem in the field, which is why I started developing the method in the first place.
My initial intent was to publish the technique and present it as a solution to the problem all in the same paper. However, after writing it up, I have noticed that it is more general than I expected and that it can be useful to tackle other problems in the field. I have then decided to write the paper entirely on it and how it can be extrapolated to solve other issues. The manuscript is quite long and I and my advisor think that the manuscript would become overly complicated if the solution to the major problem was included (some modification of this methodology would be needed, plus additional discussion).
Therefore, I am currently preparing a second manuscript which adapts the methodology to the major problem and demonstrates that it can effectively solve it. I expect this second manuscript to have a much larger impact than the first one.
However, I am concerned that the reviewers of the first manuscript will detect that the methodology can be used to solve the major issue (it is really obvious that it was designed for that, plus I suggest that it can in the conclusion) and suggest that it be included in the manuscript. While I could write a rebuttal disclosing that we are submitting their suggestion elsewhere, I am concerned that they won't accept that for an answer.
While we have considered submitting the papers as Part I and Part II, we wanted to submit the second paper to a more prestigious venue, given that we believe the second one has a much greater chance of being accepted.
What are my options? Should I submit the papers to the same venue (i.e., Part I and Part II) and/or cram everything into a single paper? Am I overthinking this issue?