If you are so close, why not just do it?
Probably suring-up support and hoping the threat will be enough to get the documents. If the House Republicans move to impeach, the Democrats will be celebrating up to the point they find out who would be impeached. Just seeing the look on their faces at that moment would make impeaching Rosenstein worth it.
Because I believe it takes something like a 2/3rds(not positive) vote to impeach. The democrats would never go along.