What is the history of the word "lobby"?
The entry for lobby in the Online Etymology Dictionary is:
lobby (n.) 1530s, "cloister, covered walk," from Medieval Latin laubia, lobia "covered walk in a monastery," from a Germanic source (cf. Old High German louba "hall, roof;" see lodge (n.)). Meaning "large entrance hall in a public building" is from 1590s. Political sense of "those who seek to influence legislation" is attested by 1790s in American English, in reference to the custom of influence-seekers gathering in large entrance-halls outside legislative chambers.
So, to answer your question, yes, it could be used in 1890s Georgia to refer to seeking to influence legislation. If you mean for it to be the entrance to a public building, then it is much older than that.
The oldest still-extant sense of the noun lobby is, per the OED:
A passage or corridor connected with one or more apartments in a building, or attached to a large hall, theatre, or the like; often used as a waiting-place or ante-room.
The OED’s first citation for that sense is from Shakespeare, and it continues to be used in those ways in contemporary English.
However, the extended sense of lobby seen in “the alcohol lobby” or “the anti-pollution lobby” and meaning
a sectional interest (see interest sb. (def#4)), a business, cause, or principle supported by a group of people; the group of persons supporting such an interest.
is a wholly modern one that arose only in the second half of the 20th century. The earliest provided citation for that sense in the OED dates from 1952 in The Economist. There are later citations from other periodicals like The Listener (which ceased publication in 1991) and The Telegraph.
Given that all citations are from periodicals, one might speculate that this is a “newsy” sort of sense. It would certainly be anachronistic coming from the mouth of someone portrayed as being from the 1890s in Georgia.
On the other hand, if you are looking for lobby as a verb, the thing we talk about when speaking of lobbying organizations and meaning either of:
trans. To influence (members of a house of legislature) in the exercise of their legislative functions by frequenting the lobby. Also, to procure the passing of (a measure) through Congress by means of such influence. Also transf.
intr. To frequent the lobby of a legislative assembly for the purpose of influencing members’ votes; to solicit the votes of members.
Then those sorts of uses arose during the earlier half of the 19th century in the United States, and so would not necessarily seem out of place in your chosen milieu.