Here’s what I would like to see the Republicans do. Submit a budget with more spending cuts than ever. Call the press and announce that this is the “emergency, get everyone back to work budget to end the shutdown.” Paint it as doing all you can to get it to pass, and you hope, HOPE the guv will sign it. Make it look like you are trying as hard as you can to end the shutdown and it is all up to Gov. Dayton, now, he has the budget and the ball is in his court. If he vetos it, repeat with even more severe budget cuts.
I like your idea...start by submitting a $28B emergency budget, followed by a $26B final submittal.