The format "ExFAT" explains why some files will fail to export.
Do not write to the top level of the file hierarchy on this drive but to a subfolder.
On your Mac use a smart album to select batches of photos systematically, so you do not export any photo twice.
- File > New Smart Album
- Set the smart rule to "Date captured > is in the range > "
Now select all photos in this album together with ⌘A and use File > Export > Export .. Photos to save the photos to a folder in your home folder, than drag this folder to your external volume.
To export at the full size use this export preset:
