That's definitely not a font. It was drawn by hand, by someone not exceptionally experienced in designing letters.
I would guess that the lowercase was based on something like Century Book Condensed, and the capitals were an invention of the designer. It's very unusual to see ball terminals on an uppercase C and S, and the curved stem of the T is something you would virtually never see in a regular serif font.
I would suggest either you find a way to precisely trace the old signs, if the client wants to retain the amateur/rustic/hand-drawn look, or find a type designer to redraw it with some improvements and updates to make it look a bit more well-drawn.
Perhaps, if you got a straight on snapshot of the main sign, you could easily duplicate that style. Looks to me, that was a pan-face, but the letters are vacuum formed in addition. Will you need to do all of that, also ?? That could get expensive, unless you find one of those old shops that are setup to do these by the 100's. In fact, that was a regular type style of some of those vacuum house sign shops back in the day.