Transfer tracks from Basecamp to ARCHIVED folder on Garmin

No, archived tracks are limited to 1 track per file.  As far as I know, retrieving an archived track simply copies the file from the archived folder to the gpx folder and deletes it in the Archived folder.  If you were to retrieve your 1st100.gpx file, you might see all the tracks in the Track manager, but you would not be able to re-archive that file, only individual tracks within the file.  At least, that is my expectation.

As far as I know, there are no utilities that will export a selection of multiple tracks into individual gpx files.

If you want to try a "Hail Mary", you could try renaming a file with multiple tracks (limited to less than 10000 total track points) to Current.gpx.  Place it in the current folder and save the Current track log from the track manager.  This might create individual track files in the Gpx folder that could then be copied to Archive folder via Windows.  My concerns are:

1. It may hang your device.  If it does, remove the batteries to shut the device down.  Hook the device to your computer and delete the Current.gpx file in the Current folder on your device.  There is a real risk that things could get ugly and restoring you device to working condition may not be easy.  A factory reset may be required which would delete all of your data on the device and profile/page settings.

2.  If it works and creates individual files, it will probably rename the tracks based on date and time.

I would give serious thought before trying this.  Exporting 600 tracks one at time will be a real pain (unless you pay an offspring/grandchild/intern to do it for you); but it is the safest thing to do.