Will the JST connector supplied in the kit work with the LIDAR sensor? I know the number of pins/wires doesn't match, but will it work because we're only using certain pins?
Or is there a different cable we should look to getting?
Your kit came with a cable that does have the correct number of pins/wires (6, I think). See attached picture. If you're finding something different, then it's the wrong cable -- don't force it into the connector. Bad things are likely to ensue.
This is the connect that you'll need to cut off one end and either solder or use electrical tape to make connectors that you can use with your Arduino on the breadboard. Something that'll look like this connector on Sparkfun ( https://www.sparkfun.com/products/14425). Note that this isn't the correct connector, so don't buy it. We were not able to source this connector anywhere.