I remember seeing a picture of Bing performing with Bob Wills at a war bond rally (possibly in Bing or Bingang magazine), and was surprised to learn that a recording of their performance of "San Antontio Rose" was made and auctioned off.
Here are an article from a Tulsa paper, and blog post from a Western swing forum discussing it:

http://www.tulsapeople.com/Tulsa-People/March-2013/Bing-and-Bob/

http://www.texasplayboys.net/_disc8/0000004b.htm

At the time, Bing had a big hit with "San Antonio Rose", which was written and originally recorded by Wills.

This would be quite a find if it showed up. Considering it seems to have been kept with some care at one point, it is a shame it is missing. This would be the one of Bing's lost recordings I would most like to hear.

I don't think it is anywhere nearly as unusual a coupling in Bing's career as his duet with David Bowie. In fact, as a fan of both Bing and Bob Wills, I always thought they would have sounded good together. Bing was a major influence on Western swing singers, including Tommy Duncan, who sang "San Antonio Rose" on Bob Wills recording. Bing's friends Joe Venuti and Eddie Lang were major influences on Western swing as well. Bing generally responded well to country material, and on Decca, he frequently recorded with a wide variety of popular and jazz artists, and generally found common ground.

I'm curious whether they used Wills's arrangement, and whether they slowed it down to the tempo that Bing had used. And whether Wills made his usual interjections during Bing's performance! I have to confess, I prefer Wills's original recording to Bing's, as the somewhat faster tempo, and mariachi style trumpets add a suitable Spanish flavor to the song.