I never heard the song; I only heard someone describe it to me. It opens with the sound of a buzzing fly, then a “smack” noise, then a female saying “Did you get him?”, then a guy replying, “No, but I will”. I believe the song was from the fifties or sixties. Ring a bell?
Shoo Fly Pie? 
No buzzing flies in the versions that I’ve heard.
Nothing? Come on…
If you don’t get an answer here, try asking on the usenet newsgroup rec.music.dementia.
Did the lyrics/music start after the spoken words? You say 50’s, 60’s, which makes me think do-wop. There were a ton of no-hit-wonder groups, some of which went for comedy in their songs. One comes to mind…I had to dig this out of my vault…ROACHES by the COURT JESTERS.
Bass voice: ‘Don’t leave your food on the table!’
Falsetto voice: Why, honey, why?’
Bass: …because of those roaches…
And the rest of the group joins in for classic do-wop harmony.
The worst part, for you, is that it’s going to turn out to be so obscure, maybe a flip side of another obscure record, that you’ll never get an answer. If you ever come up with a title or artist, I can cross reference it for you. Good luck.
You may be on to something. When my friend described it, he said “Did you get him?” in a falsetto and “No, but I will” in a deep voice. Thanks…I’ll keep after it.