You're thought process is right. Not sure why you're running into the issue. Maybe buy a bigger one yet or even better a 250 or 500GB or whatever size external hard drive. They're just as cheap anymore for a ton of space. (pictures, videos, SMS backup, etc) 

You're just copy/paste? Drag and drop? Or are you in SMS and actually trying to save the backup when it's creating it to the USB? Don't do that last option. Save the backup to your computer...or maybe it's already there. Then just copy/paste or whatever from there to your USB. Writing directly to the USB isn't the fastest or best route historically for SMS