I thought I'd drag this discussion back up from the grave, what with the new Batman movie being out and all....
I wonder what Lovecraft would have made of Bats. Maybe he'd think he was a Nightgaunt...
I seriously doubt that Lovecraft would have thought much of Batman at all. First off I can't see Lovecraft being at all impressed by the medium of comics. He probably would have thought them utterly inferior to prose literature and only something for the illiterate.
Had he lived to see Batman movie into the realm of film serials I'm sure he wouldn't have been impressed by those either. In a letter to Donald Wandrei, Lovecraft gives his opinions of some contemporary horror movies he's seen. Amongst them is The Bat
(1926). This film was actually a big influence on Batman creator Bob Kane and involves a man dressing up like a bat and also using a bat signal, the main difference being that he's a criminal rather than a hero. Anyway, Lovecraft says the film bored him to tears, so yeah I find it unlikely he would have been interested in The Batman either.
Probably his greatest Batman work is Gotham by Gaslight. It's an Elseworlds book (takes place outside regular canon) where Batman operates in the late 1890's and is persuing Jack the Ripper, who has fled England for America. Incredible one shot book. I highly recommend it.
Actually Gotham by Gaslight
is the first Elseworlds book, written before DC even had the Elseworlds label. Mignola really invented the whole concept with Gotham by Gaslight
. And yes I recommend the book too.
Also check out the Batman: Brave and the Bold
animated series episode "Night of the Demon" which uses elements of Gotham by Gaslight
(Victorian era Batman vs. Jack the Ripper) and throws in Sherlock Holmes and Etrigan the Demon for good measure.