Hello, Helen,
Don't worry about Downtown Disney not being near the Beach Club's Beaches and Cream Soda Shop (which is where you want to go rather than the Boardwalk Inn for the Kitchen Sink). It is actually easier to go from Downtown Disney than from Port Orleans French Quarter. You'll simply need to take a bus from Downtown Disney directly to the Beach Club Resort. To return home from the Beach Club Resort, you could take a boat to Disney's Hollywood Studios then transfer to a Port Orleans French Quarter bus.
If you'd prefer to go from your resort, you'll want to use any of the theme parks as a hub. For example, if you go to Disney's Hollywood Studios, you can then transfer to a boat that goes to the Beach Club.
One difference with visiting Downtown Disney is that you can set your own pace; it doesn't have the feel of the theme parks with needing to get to the next attraction as quickly as possible. You can move as slowly as you'd like and not push your family. You may want to plan on visiting Beaches and Cream Soda Shop during the mid-afternoon since it may not be as busy as an official meal time and to make a nice break in the day. If you find your children are too tired to go elsewhere, there is a
Ghirardelli Ice Cream and Chocolate Shop as well as a
Haagen-Daz ice cream store at Downtown Disney.
Have a great trip and meanwhile enjoy this
video of a Disney cast member seeing if he can take on the Kitchen Sink Challenge!