I don't know a lot about the Corolla, but I do have once owned a Nissan Datsun 1200 (yes, 1200cc) with a three-speed auto without TC lock up. It was produced in circa 1970. I got about 34 mpg (Imp), translating to about 27 mpg (US), with about 30/70 city/sub-urban driving (no highway where I lived then). I was FE conscious but did not hypermile. The car must have benefited from its own light weight.
I have had some chances in the 1980s to ride in a Corolla with a
TWO*-speed auto transmission
. It was driven by my friend. Overall, it was a very practical car for city driving. But sorry, I have never asked to know about its fuel economy. I would intuitively say that it shouldn't have been terribly bad. Otherwise my friend would have complained a lot.
Back to the question of whether a three-speed car is acceptable, I guess you can trade some FE for the Corolla's known reliability. Forgive me for stating the obvious, a reliable car uses up less of your money for repairs so that you can spend a little more money on fuel. It may not be very green, as are other cars produced several years ago, but a 1200 to 1800 cc car will quite likely be greener than a huge SUV any way.
Best of luck!
* Not a typo.