Answering my own question, using Font Forge to examine this font more closely, the Postscript name contains no spaces, as follows:
CoreSlabM27CnExtraLight
Using this name in lilypond 2.19.48 works. Since this is already an italic face, there is no need to put Italic after the name.
But the more subtle question remains: why must we use the Postscript name for this font when the so-called 'Name for Humans' (i.e. the name with spaces, to make it readable) works for all the other Open Type fonts I that I use.
Is this a bug relating to digits in the name? Should this be on the bug list? Is this a known issue? Having spent hours on this just to get a new font going, it would seem to be a usability issue at the very least, and the solution to me is not at all obvious, given other OTF fonts work fine without having to use a different type of name.
Andrew