Just an idea
If the Democrats wanted to win a ton of support right now, they should pass a law in the House of Representatives that says, whilst there is an unapproved budget that causes any government staff to be furloughed, all House of Representatives, Senate and White House members, staff, aides, etc. cannot be paid their government salaries.
If it fails in the Senate, the uproar at the Republican Senators being unwilling to feel the pain of the 800,000 government workers currently feeling pain, would drive a lot of people away from them and the Democrats would now be the champions of the people.
If it passes in the Senate, and Trump blocks it, then Trump would no longer be the man of the people, and the Republicans would need to distance themselves from him.
And if Trump actually passed it, then I can guarantee that their would be a much greater desire to find a compromise solution. After all, it would be costing all those Senators and Congressmen/women money. And we cant have that now can we?