Could it be your mother had a '68 Impala with a 307? 1968 was the first year the 307s were available.
I really wouldn't doubt it. As a kid (16-17 years old) I fell in love with the '69 Z28. I kept debating with her about the fact that they wouldn't have put that motor in the Impala. Her friend called me on it, and he explained everything about the 302 to me. Bore, stroke, compression ratio, cam specs. I only knew it was a high revving motor, and good for road racing. At which time he told me that her '68 DEFINITELY had the 302. He obviously knew a lot more than me, so I went with it. She had the '68 and before that a '63 SS.
If she was still alive, I could ask her. She actually left the car with my Father when she left him. My family always told me that he wanted that car because it had the 302.
But I've come to find out that a lot of things I was told weren't always the truth!
Oh, and if you're gonna change the gears anyway, then sure, put the 327 in there. But for the cost of having to refresh it, I personally would sell it, and build the 350, as well as change the gears.
You don't need to race to have a fun car. My '70 hearse weighs in around 7k. It's got the 472 in it. 525 foot pounds at my foot! It's a fun car to drive. My '88 Firebird Formula has a well built 383 and a lot of suspension work. My '73 Nova has a mildly built 350, and everything else is stock. PEG-LEGGIN'!!!! My '88 CRX has a 1.6l ZC motor in it. With the intake, header and muffler, I'm probably somewhere around 120hp to the fly wheel! Sure, it's not the fastest car, but it gets me through traffic. But when I want to take it up to the mountains, it holds it's own. All of my cars are fun
cars, but I've only raced two of them.
I remember working on cop cars at my last job. The box Caprices (like yours) were really nice cars. You can drive all day long and still feel comfortable. And the 350 TBI had just enough power to have LOADS of fun! It wasn't neck breaking torque, or have 0-60 times in the 4, 5 or even 6 second range, but they were fun cars! The thing about them is they had a good amount of torque at low RPMs, which makes for a fun car.
Anyone who's owned a big block can tell you that. Low RPM torque = FUN!!