I know that "past due" stamped on a bill is accepted, however I believe it should be "passed due". Does this mean that "past due" is vernacularly correct and "passed due" is grammatically correct?
'past' means something has happened, it's in the past. like a 'past exam paper.'
Passed means to either pass the exam (get a good mark) or to physically pass something when travelling, "We passed the landmarks rather quickly.'
"Past due' would mean, I think that the bill is past the date it was due to be paid.
To pass is the verb to describe what the bill did when it went beyond its due date (and what the debitor did in relation to that date in not paying, and the creditor in not being paid). So I suppose you could make an argument for "passed due", but I don't see this being a normal sense. Certainly, it would make no sense to say that "the bill is passed due", only that "the bill has passed due". We tend to label things with adjectives describing them more often than verbs describing what they have done.
Past is an adjective that is used of something that once held a particular quality or was once an example of a particular thing, but now is not. Once the bill was due, now it is "past due". This seems much more reasonable.
You could make a better argument still in terms of past the preposition meaning "beyond in time or space", since it is beyond due.
Both the adjective and preposition sense allow us to say that "the bill is past due". They also allow us to say that the bill is more past due, or really past due (as opposed to having passed due, which it either has done or hasn't done).
With the preposition sense, we can say that it's 2 weeks past due, and that penalties can accrue if it is more than 30 days past due. We can warn that if it goes unpaid until it is 3 months past due we will contact the Sheriff’s Office and arrange for bailiffs to deal with the matter.
In all, past due works well, while passed due does not.
You commonly get stickers on, say, electrical equipment, saying Passed, where the meaning is something like Passed Inspection, or Passed by Quality Control.
That sense isn't really applicable with overdue invoices, because they're automatically overdue if still unpaid after some particular date which will always have been either implied or explicitly specified when the bill was first issued. There's no need for someone in the accounts department to actually examine the account and say "I will pass this bill as overdue".
So although OP could make a case out for saying Passed Due would be "credible", that's all it is - a potentially credible case that didn't win the argument when the stamp-makers were deciding what wording to go for. Thus, the stamp wording stands in for...
it [the current date] is now Past the date on which this invoice was Due to be paid.
[Not] the arrow of [current] time has Passed the date on which this invoice was Due to be paid.
[Not] our accounts personnel have Passed this invoice as still unpaid after its Due date.
In short, OP's "peeve" about the choice of wording does have some substance - but if the stamps had gone for the alternative Passed Due he could just as well have peeved about that.