While partial to anything Rat Pack-related, I've never seen this 1971 sendup of westerns, complete with Bacharach theme song.
Dean Martin stars alongside a stellar supporting cast, including Brian Keith, Ben Johnson, Honor Blackman, Albert Salmi - and especially the crappy little dog we see here at 0:08.
Looks like campy fun, and if this features anything other than Dino consuming mixed drinks with showgirls at the Cal-Neva Corral, I'd be very disappointed.
Since Mel Brooks soon ended up featuring a gag in which Count Basie And His Orchestra perform in the middle of the prairie (in Blazing Saddles), it's a good bet that we don't see Burt Bacharach whooping it up on the ivories at the Lone Star Saloon in this film. . . but I could be wrong.