In the current case, President Trump could break the impasse over the Court by appointing Federal Circuit Court Judge Amy Coney Barrett (or some other qualified person) to the Court via a recess appointment to last until January of next year. He could say, in taking such a step, that his intention will be to make that appointment permanent in 2021 should he win re-election in November. That would turn the appointment into an issue that might work to his advantage in the campaign, while avoiding a nomination that would divide the Senate and his party. The recess appointment would be temporary, lasting just a few months, and would still allow the next president to fill the vacancy on a permanent basis in January. It would also fill a vacancy on the Court, giving it a full complement of nine members when it starts its next session in a few weeks.
Senator McConnell could assist in such a step by declaring the Senate to be in recess for a period in September or October, thus allowing the President an opening for a recess appointment. That would not be difficult to do, since the Senate is likely to be in recess anyway in October to permit members to campaign for re-election.
There are risks to such a move, but there are risks both in trying to fill the vacancy or in leaving it open for the time being. It would provide an opening for new attacks on President Trump, but those attacks might drive other issues off the national agenda for a time while permitting the President to maneuver the issue in his favor. Assessing the risks, a recess appointment looks to be a reasonable way for the President to fill the vacancy in the immediate term while still allowing the next president to fill it on a permanent basis.