This is what I am doing all the times:
- sync the image folders (images and xmp) between home and mobile computer
- sync the mentioned profile folder between the two
- optional: sync the cache folder between the two
In your case:
- step 1 may be obsolete as you have those files on an external drive
- but then: the path to the images on the external drive must be the same on both letters (in windows: same drive letter etc.)
… and as you mentioned … exact same version of dt on both computers …