How to Present an (Essentially) The Same Proof, in a Different, Earlier Working Paper, in a Present Paper?
Presumably you cited the earlier work. If so, I don't really understand the concern. But, if the editor/reviewer/conference would accept it, you could offer to show them the proof separately from your new paper. This would permit them to verify the result if they think it necessary. 

But you were correct in not including the result in the new paper. But that also assumes that the earlier paper will actually be published. And it is possible that it won't be. 

However, there is an alternative, but it would also require some editor assistance. For purpose of review, you could include the proof but note that it would need to be replaced by a citation if the earlier paper gets accepted. The final, published, version would be a bit different from what was reviewed, but that is normally the case anyway. The easy way to include the proof would be in an appendix.