In 1923, WTAX-AM was founded in Streator, Illinois by the Williams Hardware Company. they began with only 50 watts on 1210 AM. Today it operates at 1,000 watts. In 1930 Jay Johnson moved WTAX to Springfield, housing it's studios in the Abraham Lincoln Hotel. That was the missing link.
At that time, they shared the 1210 position on the dial with WCBS-AM. In 1941 that was rectified, and the FCC approved a move to 1240 AM. In 1943 Pittsburgh Post-Gazette editor Oliver Keller bought the station and they became a CBS affiliate. Currently it's owned by Saga Communications. They no longer occupy the Abraham Lincoln Hotel. I can't nail down a date for that, but it appears to be in the mid 1930s.
I did find one interesting source that grazes the topic. I found an interview of an early WTAX owner, Shelby Harbison from 1973. The interviewer is Kay McGuire. Shelby actually worked selling radio time at the station while they were located in the hotel. The interviewer asks about the record. Read below, or click on the pdf. Man I want to hear this thing.
Q: Why does a radio station make a record on Lincolns funeral?
A: Well, because the radio station was owned by Oliver Keller and Keller was a historian, this was his key hobby. Keller was also a Lincoln worshiper. He knew Lincoln, studied him. Keller was very active in historical circles, the Civil War round table. Keller really revived the current interest in the Abraham Lincoln Association... Keller also, when it came around to the one-hundredth anniversary of Lincoln's burial, went back researched it and authentically wrote up this playlet...