Performing segue after sender view controller is dismissed

Based on a modification to this answer, the following should work:

In your storyboard, remove the segue that is triggered by tapping your menu button and goes to Destination.

Create a new segue that goes from the Origin view controller to Destination view controller. This segue is going to be manually performed.

When your Destination option is selected in Menu, have Menu dismiss itself and then perform the Destination segue on Origin, like this:

    // This code goes in Menu, and you should call it when
    //    the menu button is tapped.
    //
    // presentingViewController is Origin
    weak var pvc = self.presentingViewController

    self.dismiss(animated: true) {

        // Menu has been dismissed, but before it is destroyed
        //  it calls performSegue on Origin
        pvc?.performSegue(withIdentifier: "openDestination", sender: nil)
    }

When Destination is dismissed, you should see Origin, without seeing Menu at all.

I tested this in a sample app where "Menu" was not a slide out, but a full modal view controller, and it worked for me.

EDIT: While troubleshooting with @KingTim, he found that we needed to wire the segue from the UINavigationController, not Origin, to the Destination. This is because Origin is inside a navigation controller. After that discovery, it worked.

If your presenting view is embedded in a navigation controller then you can do this:

    weak var pvc:UIViewController! = self.presentingViewController?.childViewControllers[0]
    dismiss(animated: true)
    {
        pvc.performSegue(withIdentifier: "SegueID", sender: nil)
    }