Ok I've done a bit of scratching around the interwebs and think I have a few answers. First up is the registration information of the Pin-ap-ola brand from January 1916.
The home offices of the parent company are located in New York City.
Chris Weide's page on the local bottling companies of Jacksonville, Florida gives us an idea of the connection between Pin-ap-ola and Pin-ap'a.
The Pin-Ap-Ola Bottling Company of Florida begins operations on April 24th, 1915. They change their name to the Pin-Ap-A Bottling Company of Florida on May 17, 1916 (a few months after getting their trademark registered). The company still exists as the Pin-ap-a company into 1917, after that nothing.
My guess is that just like with Gay-Ola, Coca-Cola quickly pounced on the Pin-ap-ola parent company when it registered its trademark and forced a name change for the product.
A. L. Blanhord, Southern representative of the Pine'ap'ola
Company, of New York City, who has been in Mobile for the
past three weeks, has announced that his company has made
arrangements with a local concern to erect in Mobile a $20,000
soda plant for the purpose of turning out their products for
Alabama and Mississippi. The Pine'ap'ola Company was incor-
porated under the laws of the State of New York. J. J. Warren,
the famous sugar king of Havana , Cuba, is the president of the
company.
Madman per our conversation last night: If you will notice the Lamont, Corliss & Company American distributors notation appears to be an integral part of the ad, which leads me to believe that not only has the name been changed, but it is possible that the brand has been sold to Lamont, Corliss & Company. While the Tenn-Cola Bottling Works & Ice Company notation appears to be something added. That is because the print block from the parent company would have left a cut out in the advertising block for the local distributor to place their own additional block in the ad. This isn't immediately evident on the first ad, but the second you can see smudges around the inserted block. This was a very common practice in national brand advertising at the time.
Lamont, Corliss & Company of New York has been around for a long time, it was one of the first companies to distribute Cream of Wheat, and also owns the Nestle brand.
More information from Chris Weide's website. This article gives a lot more information on the origin of the brand, the people involved, and who provides what part of the process of creating the drink.