I'd also recommend not limiting your repetoire entirely to "Celtic" songs & tunes. Numbers you like and can be adapted to your style as a "Celtic" band can really round out a set nicely. There's plenty of precedent. The House Band do Tom Waits' "Shiver Me Timbers" and Bulgarian folk tunes. Makem and Clancy did Michael Smith's "The Dutchman" and Gordon Bok's "Peter Kagan and the Wind." The Battlefield Band and the Tannahill Weavers, among others, have done Stan Rogers' "Barrett's Privateers." The Blarney Brothers do James Keelaghan's "Fires of Calais." Mary Black sings Stephen Foster's "Hard Times Come Again No More."
The Buckaroos (the loose-knit band I'm in), both in "Celtic"-mode gigs and non, do Sydney Carter's "Lord of the Dance." Oh, and "The Dutchman," for good measure. :-)
If you feel like it, find things like this that you like and want to play. Audiences may be surprised and delighted to hear songs they know from another setting in your style.