Games : Is the game unwinnable if you don't save the mouse?

For all the flak that King's Quest V gets for the custard pie puzzle, I got stuck on something that seemed far worse to me: the mouse. When I played, the following happened:

1. While walking past the bakehouse, the narrator started talking.


2. I had accidentally clicked through what he said and suddenly a cat ran out and, in about two seconds, grabbed the mouse.


3. Figuring this was just a random background event, I went on with my game.


4. I got past the poi -sonous snake! and went to the mountains.


5. When I reached the frozen waterfall, I found I couldn't advance.

Frustrated by this, I read some walkthroughs, which revealed that:

This seemed like a really bizarre sequence of events (at least the custard pie didn't require you to do something you knew had been suicidal in the past in the hopes that someone would save you from your stupidity) and sadly I had to backtrack a couple of hours in order to get to a point where I could save the mouse.

It's made me wonder: do you have to save the mouse and start the sequence of events to get the inventory item you need? Or is there some other way to get past the frozen waterfall?