While at the library (not Bridgeport), used Genealogybank and found two small items in a New Haven & Hartford newspapers, The Hartford Daily Courant dated 6 July 1871 under State Correspondence - Bridgeport: Patrick Flynn committed suicide Saturday evening ... although several persons were in the house at the time, no one was aware that he made an attempt upon his life until some time after the deed was done. He lived about six hours after inflicting the fatal wound.
The Columbian Register from New Haven on 15 July 1871 published the following: Patrick Flynn living on Hurd avenue Bridgeport committed suicide Sunday morning after having been on a spree several days. Note: this newspaper might not have been a daily & news under "correspondence or items" were often published days or weeks after the event.
Church records on a suicide might be scarce - burial might not be in the Catholic cemetery but in the "Catholic" section of a secular cemetery.