If I were in your shoes, I would:
1. Export the RAW file to a TIF, in the same folder as the raw
2. Copy the TIF to an external harddrive
3. Attach the HD to the PC
4. Process the image, saving the updated image into the same file on the HD
5. Attach the HD to the laptop
6. Copy the TIF from the HD to the original location on the laptop
7. Open LR in the laptop
8. Read the metadata from the file to the catalog.
Alternatively, you could move the catalog and all of the images to the external HD on the laptop, the do steps 3, 4, 5, 7, and 8.
Alan