Is it all caps?
If so, try Augustea, Trajan, Goudy Forum, Ceasario
If not, try Goudy Old Style, Centaur, Clarendon.

I haven't decided yet, but probably not. It's a larger plaque than strictly necessary, so it will include extra historical information and a black silhouette of the structure. It's still up in the air. Thanks for your suggestions. I'll take a look!

What was/is the building used for?

It's the Western Newspaper Union building. I understand it worked something like an AP service. They would create inserts or plates here and ship them to of independent presses across the country.

From the registration documents:

"The Western Newspaper Union Building [preferred] at 304 W. 10th Street, Kansas City, Jackson County, Missouri is eligible for listing in the National Register of Historic Places under Criterion A for its local significance in the area of COMMUNICATION. In 1901 the building at 304 W. 10th Street was constructed for the A. N. Kellogg Auxiliary Newspaper Company to house their printing operation that sold pre-printed auxiliary newspapers to over fifty-five (55) small town newspaper and quarto publishers in a three state area...

The early development in the auxiliary or ready-print newspaper (also known as boiler-plated newspaper) industry in the United States began just prior to the Civil War c. 1860. Several small printing firms in the northern portion of United States began following the work of Cassell, a London based publisher (c. 1857), who supplied approximately 150 newspapers with ready-printed sheets of national news and advertisement in England. The operation in England remained a small scale process compared to the American adaptation in which the process was expanded to become the largest systematized news gathering and distribution method in the world."

If you're interested just google "Western Newspaper Union Kansas City" and the PDF is in the first SERP. It's too large to post here, else I would do it for you. It's now being used for a technology services company, and for apartments on the second floor. There were no press plates, type samples, or anything of interest when we bought the place. I scoured every inch of this place.

Maybe, as @Andreas_Stiftung suggested, a Clarendon would do you the best?

You stated you wanted the sign to be clean and readable with some sense of the era. I can well imagine an American Gothic in use there. I really like the revival of Alternate Gothic but my first pick to try out (if it should be clean and more modern) would be Tal Leming’s Balto.

If you are looking for a newspaperish text flavour, try a Scotch. Something from the Century type family or a contemporary one like Chronicle could work.

For additional research on your own, flip through digitized books of American Type Founders of that era, like this specimen from 1912. (Oh, and if you need a display or headline typeface resembling the octagonal “Western Newspaper Union” lettering from that photo you posted, give one of the narrower styles of Winner by Christoph Koeberlin a try. There is also a sans-serif upcoming but I might prefer a serifed version.)