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:

You have to throw the shoe at the cat before it catches the mouse. The mouse thanks you and says it will return the favor. Then you have to go to the inn, despite knowing full well from the last time you were there that the thugs will club you and give you a game over. You get tied up in the cellar and instead of the game over, the mouse shows up to chew through your rope, allowing you to escape. Now you have a rope you can use to cross the frozen waterfall.

This seemed like a really bizarre sequence of Sierra Logic (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?