Is there research on the extent to which complexity of the proposed method(s) impacts the acceptance rate?
I haven't seen any research on the correlation between the complexity of the methods and the acceptance odds of a work and I think it is relatively hard to do a study on this. First because being simple or complex is completely subjective and domain dependent and hard to quantify. One can not generalize a pattern that is seen in a particular scientific community to others. Moreover, the information about rejection of papers is not publicly or easily accessible making the hypothesis hard to investigate.

That being said, in computer science machine learning community, I have seen that people are pushing towards critiquing the unnecessary complexity of the suggested methods and putting more emphasis on the actual results rather than the simplicity or complexity of the methods. There are many simple methods that are being published because they actually work well. I personally don't think that simple ideas which show good results are necessarily prone to rejection, but the opposite argument (complicated methods having higher chance of acceptance) is sometimes true.

On a related note, explanation of the ideas and presenting a work in the most simple way for people to understand, is being emphasized more and more. One reason of people trying to complicate things is that they think the less it is easy for reviewers to understand their work, the higher their chance of acceptance. I personally think that ideas in the papers should be explained simply and clearly because if it can’t be explained simply, probably it can be done better. And reviewers should not praise the work they don't understand, because it is the duty of the writer to explain everything simply enough so that the corresponding community understands it well.