I don't have anything else to suggest.
The thing is, either it's a chemical staining, or it's an organic one.
Chemical is probably something to do with the material the lid is made from, while an organic one is a 'stain', and either it'll scrub off...and didn't....or it needs to be bleached or 'biologically' as in washing liquid type biological stuff used. Vinegar didn't work so that chemical option's off the board.....kind of leaves bleach really, because biological washing liquid always leaves a taint of soapy perfume, while bleach does eventually dissipate.
You could try salt, but salt can be very damaging, moreso than a short bleaching.
Sorry not to be of more help.
M