#11 It is usually followed by a noun: -- by way of apology / explanation / introduction / contrast, etc.
True. But "escaping," as a gerund is grammatically speaking a noun. The same sense would be conveyed less elegantly but syntactically correctly by "like someone asking to have a look at the roses by way of escape from a deadly tea conversation."
Hmmm. Maybe Highsmith's use of it is unusual but I nevertheless see it as exemplifying precisely the usage Hecuba - UK cites in the last line of #11. It's different insofar as Highsmith follows it with a gerund, but there's nothing wrong with that by my book.
(I have to admit, though, that the first sense of the phrase that came to mind when I read the OP was the other sense of it,* which made no sense, and after a while there was a penny-drop moment.)
* And this as a result of its being followed by a verb form, which one initially reads as a participle and not as a gerund, rather than by a self-evident noun.