Hi Chris,
ImageIntegration.integrationImageId is a read-only output parameter. It is the identifier of the integrated output image after the instance has been executed, but you can't change it from a script or otherwise.
ImageIntegration does not write images to disk. You can write a relatively simple script to write the integrated image to a disk file of your choice. For example, this is what the BatchPreprocessing script does to write master bias, dark and flat frames. Let me know if you are interested in knowing how to do this.
You can also save your entire PixInsight session to a project file. This is normally the best option to store a complete work including all the stages from preprocessing (calibration/registration/integration) to postprocessing.