Here's the Carlisle Brothers' version of "Rainbow At Midnight". Do I hear "The Tennessee Waltz" in it, yet to be composed by Pee Wee King (with words by Redd Stewart) "close to Christmas in 1946" according to Wikipedia?
The Carlisle Brothers' recording of "Rainbow At Midnight", although virtually ignored by the Billboard reviewer in behalf of the B-side "Live And Learn", made it to #5 on the Juke Box Folk chart (week ending Oct 18, 1946, miscredited to Bill Carlisle only), and Ernest Tubb later had his third #1 on the Juke Box Folk chart with his version of the song.
According to Praguefrank the instrumentation is just Bill Carlisle on rhythm guitar and Cliff Carlisle on steel guitar and both are on vocals, but the A-side recording sounds more like it reads on the label: "Vocal Trio with String Band", with apparently a wind instrument (a clarinet?) added at least.
The same (vocal trio with string band) seems to be true for the B-side that I have not heard yet, see Billboard review: "There's little of interest in the waltz, but a considerable amount of outdoor appeal is lodged in Cliff Carlisle's Live and Learn, here given voice to by the trio. A steel guitar kicks things off on this side, the string band plays along behind the boys and a dancey beat is maintained."
Maybe someone (Mike Gann?) can help out with more details of the session.