There certainly is a similarity of names - even beyond your story, but I do not see a clear family connection just yet.
The similarity in names arises from the fact that my grandfather, Samuel Schwartz, born in Monasterzyska, married Dora Kessler. She was from Pruzhany, Belarus. They met in NYC and married in Toronto.
Moshe was born in 1815 and died in 1900. The oral history written down in the family is that he was born and remained in Monasteryzska. He made rope. We do not know of any connection between him and Stanislawow.
My great grandfather, Peretz, lived for a while in Stanislawow, possibly sometime not long before WWI and definitely after the war. He may have died there. His home was shelled during the war and he and my great grandmother survived in the basement. I do not have good dates for any of this.
It is the family's understanding that all of Moshe's children were born in Monasterzyska. The earliest connection to Stanislawow I have found is with Moshe's son, Mordachai, better known as Markus. Mordachai married at age 25 Chale Sure Ebert in 1880. I presume Mordachai moved to the Stanislawow for business not long before 1880, met Chale and they married. They had 15 children. Mordachai was my grandfather's uncle, and most of Mordachai's children emigrated to the US.
I am in the process of identifying all Schwartzes (including spelling variations) from Monasterzyska via stevemorse.com and will chart them and their families from info in the manifests or possibly naturalization documents. I'll try to do the same for Stanislawow-associated Schwartzes per stevemorse, I'll share this info with you when I pull it together. I'm also planning on posting to Viewmate some 20th Century Monasterzyska-associated Schwartz metrical records for translation in the next week for inclusion in the "chart."