fan
I found a website titled "Gundata" which states that the .270 WAS offered carbine form from 1960-1963. So too the .270, which went a year longer till 1964.
That aligns with my memory of the rifles from the mid 70's as a kid. I was a fan of the 760 pump, and especially wanted a carbine. I suppose, in those pre internet days of the early 70's, I looked at Remington catalogs more than I watched TV. (we only had two channels!). I cannot recall the Carbine being offered in anything but .308 and .30-06. Eventually, they dropped the .308.
Gun Data says the last offering of the .308 in carbine form was 1980. I still haven't gotten a .308 carbine pump!
It wouldn't be original, but it would be a simple matter to have the barrel of a .270 pump professionally cut and recrowned, and the front sight reinstalled. But I do not believe the .270 is a good caliber for short barrel conversion, or the '06 for that matter either, but it could be done. There was a recognized family in the NE (Benoit's) that cut 760 pumps in '06 and were pretty famous as deer trackers.