D2 ~ (How's R2 doin' these days?)
There is no connection between "Chaser" and "7up" that I know of. I am 99.9% sure that one will not be found ... other than they were both Alcaline/lithiated beverages. There were numerous lithiated-type beverages back in the day, (they were a popular drink) and many were copycats and/or knock-offs of others.
But I do know there are variations of the "Chaser" bottle, (see below) and that the one you have is considered extremely rare, and possibly quite valuable to the right collector.
The one like yours with the embossment on the shoulder is the earliest one from that brand, and rarely found in good condition. Later they changed it to "Chases Thirst" instead of "Chases Hangovers." Plus they eventually replaced the little "devils/demons" with a greyhound dog to suggest the "chasing" aspect. They were made in both Chattanooga, as well as in Memphis, Tennessee.
I would be real interested in knowing the date on the base. It will most likely be a two-digit number, and from the 1930s. Look for a 38 - 39, etc. Or it might look something like this ... 21 <(I)> 8 ... for 1938
The earliest 7up bottle (also shown below) is dated 1935. It is one of only a very few acl (painted label) soda bottles from that year. The acl process wasn't introduced until around 1934. As yet I have never seen a painted label soda bottle from 1934. The only 1934 acl I know of is a milk bottle by "Kolb's" dairy.
I hope this helps. And please let us know if you intend to sell your "Chaser." I am certain you can get top dollar for it, and will speculate to say at least $100.00, and possibly even more.
SBB