The function that the live editor uses to exectute a livescript is the following:
Note that filePath is the full path to the mlx file, otherwise it will return an error. If you run it in the command window, you can then interrupt the execution of the live script by using Ctrl + C, as usual. If you find the feature useful, you can program a shortcut script for it.
However, executeAndSave, as the name suggests, not only executes the mlx file, but also saves the resulting code output and figures, meaning that interrupting it can cause some unexpected errors (I have encountered one during my few minutes of testing, so they must be quite frequent), and you have to close the Matlab process entirely. Which is why I suppose the feature to interrupt live script execution wasn't added in the first place.
This answer probably won't solve your issue, but you are now at least more aware of the latent reason for it. Good luck!