Geography Challenge: European Watershed

Bernese here. If I recall correctly, all of the water, that flows through Berne ends up in Aare, Rhine, Rotterdam and eventually the North Sea.

Today, I stumbled upon this visualization of the european watershed and realized that a big part od Switzerland is sandwiched between rivers that either flow to the Mediterranean (Rhone, Po) or the Black Sea (Donaube). This made me think: How far do I need to travel from the city of Berne to touch a River, that does not flow to the North Sea?

Rules:

• Find the shortest route
• Starting point: Cartographic reference point of Switzerland according to CH1903 in Berne: https://maps.app.goo.gl/qS59Lp1519YAaqf98
• Goal: River that does not end up in the North Sea
• The exact place needs to be accessible by car, bike or foot. But not a hour long alpine trek.
• Post a route and a short explanation.

Here are my best shots so far:

• Veveyse: 75.5 km: This river flows into Lake Geneva (Lac Léman) and the into the Rhône and the Mediterranean
• Doubs: 79.7 km river on the border to France. The water goes to Saône, Rhône into the Mediterranean.
• Lonza: 87.0 km: The Lonza river also merges with the Rhône in Gampel, Valais and ends up in the Mediterranean as well.

Could there be any better spots?