Donald Trump may push Senate Republicans to try to jam a Supreme Court nominee through before the election, but I think it's more likely that he'll opt to run on the vacancy given that it's an issue that could bring Republicans who don't like him back into the fold. It would be better for him than running against the Democratic backlash that would follow a hasty confirmation before the election. And Senate Majority Mitch McConnell would also be hard-pressed to usher through a confirmation in that brief period, and he has vulnerable members who need to be home campaigning.
Sens. Susan Collins (R-ME) and Lisa Murkowski (I-AK) have said that they will not vote for a nominee before next year's inauguration. Mitt Romney (R-UT) was reportedly against moving a nominee this year as well, although his press secretary denied the accuracy of the story. If he's a no, then one more vote kills a confirmation, which would be a devastating blow to Trump just before an election.
That makes it likely that Republicans move during the lame duck session between the election and a new Congress being sworn in in early January. If Trump wins a second term, then it doesn't matter. If Biden wins but Republicans hold the Senate, then in all likelihood, McConnell will rush a Trump pick onto the Court.
Democrats would have a powerful argument about respecting the will of the voters if they win the Senate and the White House, but Republicans tend to be unmoved by majoritarian appeals. They would only have real leverage if they win unified control and can threaten to get rid of the filibuster and expand the Court (or enact other deep, structural reforms).
As if the 2020 election weren't already stressful enough, in all likelihood it will determine the future of the Court, and with it efforts to combat climate change, expand public healthcare and virtually everything else on the Democratic agenda.