What is the origin of "Judas gate"?

While reading the Jack Higgins novel “The Eagle Has Landed” (1975) I came across the phrase “Judas gate”. Research on-line indicates he is rather fond of the word, going to the point of naming another book by it. His fascination with the word is discussed in a letter.

"Judas gates constantly surface in my book and people have often commented. There is a painting, I think Victorian, which shows a very large double gate. Inset in this gate, is a small door or gate which you can open and step through without the inconvenience of opening the larger double gates. In the painting, Judas is seen stepping through on his way to betray Christ. So, in English factories, you find such a small door set in most big factory gates and they were traditionally known in Yorkshire, a very industrial area where I grew up, as JUDAS GATES..."

I have not been able to identify this painting on-line, and a search of Bible texts mentioning Judas does not render any references to gates as supposedly depicted in the painting.

However, I did find the phrase in earlier books in a Google-books search going back to 1780, while ngram shows a spike in the usage of the phrase around 1981.

The closest I can find on Etymonline only has Judas goat.

If they were called Judas gates before the painting, and there is no mention of Judas going through a gate in the Bible, how did this type of entrance come to be called a Judas gate? Is there an oral tradition in England to explain the name?

The OED has one meaning of Judas being

A small aperture or lattice in a door, orig. the door of a prison cell, through which a person can look without being noticed from the other side; a spyhole, a peephole.

with the earliest quotation being from 1837: "Following the slow march of that whitish square that the Judas at my door cuts out upon the dark wall opposite to me."

Presumably the idea of an opening through which spies can look in without alerting the prisoner was extended to door through which spies can slip out without alerting those outside.

For what it is worth, I have been using the term for many years without checking the etymology: I vaguely assumed that it was a means for someone inside to betray the bulding by admitting a besieging army (or, more probably, someone out after curfew) without the noise of opening the full gate.