While I am continuing to try to learn more about the piece in my OP I thought I'd share what I've been finding. None of it is definitive but I thought worth sharing.
The obverse of the medal refers to the main building of an international exhibition held in Philadelphia. The date on the medal is 1877. Indeed, there was an international exhibition held in Philadelphia but it was in 1876 not 1877.
Also, there was a man named George Soley who had acquired the first steam powered coining press used at the US Mint. He is referred to as a die sinker and is known for having produced medals at events of this sort with the Lord's prayer on the reverse between 1860-1880 according to an excerpt on
https://en.numista.com/54974 describing one of these medals. This same medal was part of an earlier discussion in this subforum (
https://goccf.com/t/479228). However, in the details about this exhibition I found on Wikipedia there is no mention of George Soley as an exhibitor.
I have also seen pictures posted on another coin collectors message board that have the same details as my piece but one seems to be copper-like and one maybe silver. The copper-like one is dated 1877 and the silver one 1876. Even the described sizes are consistent with mine. I'm not sure if it's appropriate for me to re-post those pictures here so I won't unless it's useful and staff say it's acceptable.
I have also seen other medals of this type attributed to Soley with obverses showing various President's of the time (e.g. Grant, McKinley)
This first coining press ultimately was donated to the
ANA by Soley's widow after his death. The
ANA had plans to strike some new medals in gold, silver, and copper for their summer seminar in 2000 according to a news release (
https://www.money.org/ana-accepts-m...team-press/) . The news release suggests that these medals would be 27mm not 18 and not date is mentioned. The release also states "After his death, Soley's widow gave the press to The Franklin Institute Science Museum, which electrified it and struck medals for visitors to its museum in Philadelphia. " but does not give a date for when this took place.
Anyway, it seems like I'm all over the target but can't find the bullseye.