Very true, you would loose everything on it by Formatting.
I believe the reason it's saying "disk not formatted.", is simply because the first Sector , (Sector 0),doesn't contain that "Non-System disk... Replace and strike any key" stuff. If that stuff isn't there, the silly PC says it's not formatted.
If you have Norton's Disk Editor, open a PC Floppy and look at Sector Zero, you'd be copying that Sector, then inserting an Atari Floppy and writing that buffer to Sector zero of the Atari one... Well I do it on an Atari using Disk Doctor
Look at a few Atari vs. PC ones, maybe test on
You also might find someone you trust with your data to just read your floppies, zip them up and send them to you.