Destry:
I used my "call a friend" option, and reached out to someone that I knew had a good frame-of-reference for organized outdoor activities and the Meriden area.
His answer to me:
"My dad belonged to the Meriden R&G for a time I remember, we left there when I was 10. Silver City and Parker make sense as the Parker shotguns were first made there and International Silver was a big player in town, during the [Second World] war effort my mother was there as they were moved to make surgical instruments. There was also the Blue Trail rifle range in the nearby area, but not sure if it was in the city limits. It was an open range but I don't think it was a membership club although there may have been some leagues etc"
Not dispositive, by any stretch, but it may contribute to your being able to piece together a plausible answer to your question.
I see when looking online that there is a Blue Trail Rifle Range today, with a Wallingford, CT address. There may be someone there that could flesh-out some background detail for you. I know that some R & G clubs have retained their original place names after having moved to a region of less congestion.
https://www.bluetrailrange.com