It depends on how married you are to the idea that they must literally be the exact same mountain range. From the strictest geographic perspective, once the mountain chain is no longer in a single unbroken line, then it's considered two separate ranges anyway, even if they have the same geological origin.
Based on the first map, which is very similar to the west coast of North America, then yes, there is another process by which a similar topography can happen that would not involve the transverse fault running horizontally through your second map.
As the North American Plate moved west, it created chains of volcanic islands in front of it, called island arcs.
Eventually, the North American Plate plowed into the islands and they glued themselves to the front of it, as mountain chains. In geology, this is called accretion.
As North America has continued moving west, the former-islands-now-mountains have continued to get taller, especially in western Canada. You can get an idea of what this process would have looked like, over geologic time, here:
So, going back to your first map, what you would have would technically be two separate mountain ranges, very close together. They would have been two island arcs, very close together, that were accreted onto the main continent. This could result is a valley with a river system similar to yours between them, but the ranges could also meet somewhere off the top of the map, the way the Coastal Ranges meet up with the Sierra Nevada.
Edited to add: This way, you don't have to have a separate explanation for how the island to the south of your western mountain chain formed. Since it would be part of western island arc, it would make sense for the island to be slightly behind the eastern range. You wouldn't even need to say that it was still accreting - similar to Vancouver Island, you could have the leading edge of the continental plate be just to the west of the island and its associated range.