The problem you will have trying to match it with a single font is that the lowercase is basically like a glyphic italic* with minor swash, and the cap B and S are both swash scripts.

Take a look at a few of the script and italic faces on FontMesa.com and see if you like them (though the scripts are connected, the have the feel).

*FYI -- "glyphic italic"
Nick Shinn's Beaufort Condensed Heavy Italic has a similar "feel" in the lowercase, but a font is more precise and consistent than a handlettered sign.

Looks like the Burma Shave logo (used in this ad during WW II) is quite different. I guess the sign painters had some latitude in producing one of America's most famous roadside ad campaigns.