Based on the normal timing of a lawsuit it is very possible that the suit will backfire on the Republicans and the Democrats will get press supporting the President and complaining about the House's wasteful spending.
This assumes that the House's Council files the Complaint in August or September.
The President's Council will be able to file their Answer in October while voters are actually voting. The Answer and Motion to Dismiss will drive the news cycle up until election day.
Even if the Complaint is filed in October the President's Council will likely have the Answer and Motion to Dismiss ready to be filed quickly even though it wouldn't actually address the Complaint knowing that they could amend those filing after the election.