People make the mistake that the Monarch has some sort of independence, or can act as a last stage defence against fascism (etc.). The monarch cannot, and only acts on government advice. In other words, it is a power given to the executive.
The answer is to do as with the Fixed-term Parliaments Act. Instead of passing legislation to extend the date to avoid a no deal exit from the European Union, pass an Act of Parliament to turn proroguing Parliament into Statute Law. Then Parliament - specifically the House of Commons to do this - can enact the law in the specific case to keep parliament open. That done, there becomes ample time to extend Article 50, via another law passed.
It is said that Johnson won't present such an Act of Parliament to the Queen, and of course the Queen does as the Prime Minister says. This would then go to law, and the precedent is already set that breaking an 'embedded convention' is illegal, whereas breaking a simple convention is legal. Johnson has probably broken a convention by proroguing Parliament for such a long time, but not to deliver an Act of Parliament to the Monarch would be to frustrate democracy itself and would break an embedded convention. The Supreme Court would therefore force the government to act on pain of legal penalties.