Here's another take on the Kelly's Love variant; Daniel Hardie (2002) Exploring Early Jazz, writes about Chris Kelly's Band 1918/19:

"Kelly was particularly known for his muted playing of "Careless Love" a tune someone renamed "Kelly's Love". Some people associated with him claimed that Kelly composed the blues." (p126)

This is an interesting idea, although Hardie doesn't reference any clear sources for this information.
However, the idea that Chris Kelly was particularly known for his rendition of Careless Love is given elsewhere, for instance in Scott Yanow's (2001)Classic Jazz, but again with no reference to a source for the information.

Also in the The Rough Guide to Jazz, where Digby Fairweather writes:

"Kelly's famous feature was "Careless Love" which (says researcher Len Page) he played into a plunger-mute at a time when the technique was new; the effect of the rendition (it was claimed) made men weep and women tear their clothes off. To at least one New Orleans citizen,Kelly's wife Edna, the song was really titled "Kelly's Love", and it became to his admirers, who knew his whispered, deep-muted variations note for note." (2004: 433)

Again, not exactly comprehensive in its referencing of sources.