How can I find a fossilized spine to donate to the Island Field office on Ginger Island?

Seems like the Iridium Rod should do the trick for catching the spine out of the river according to this video. Make sure you are fishing from the river that in the dig site, and not the small one you cross to enter the dig site:

river

Alternatively, you can also find the spine from Artifact Spots on the south beach, but fishing is probably quicker.

Like the video illustrates, a Reddit post says the spine is caught similar to trash or also in a treasure chest:

Spine: fishing in the river at the digging site (EDIT: can be fished up in treasure OR in like trash/algae)