I need to convert a collection of GEOTIFFs to a format that can be uploaded to Cesium Ion with the spatial reference system intact.

What formats does Cesium accept?

Here's the list (under "Imagery"): https://cesium.com/learn/3d-tiling/tiler-data-formats/#supported-data-formats

Even though it accepts GeoTIFFs, I have over 600 of them and all together, they're too large for it to accept.

Try creating a .prj file with WKT representation of coordinate reference system.

All the files should have the same text and the filename of the .png that references it.

You could obtain the WKT by writing a BBOX of one image to .shp and using the text (of the .prj writen) in a Text File writer via the text_line_data attibute.