There was a lot of slack given to 'campaign clothing' - this would include something like the bandanna, so pick something easily available at the time from one of the Sears, Roebuck's catalog reprints and go from there.
Patterns don't matter - nor do colors - we wouldn't wear Branch of Service color around our necks until Korea, so whatever appeals to you will be fine.
Remember - it'll be 100% cotton, and it'll be 'big' - like the Army Cravat bandage @ 37"X37"X37" - that way, it was actually usable and not decorative.
Campaign clothing took a beating - make sure your bandanna looks washed out.
Scouts Out!