Powering a 5V Arduino Pro Mini with 3.3V I want to run the 5V board at 3.3V because I can get the 5V version cheaper than the 3.3V version.

I read that you can power the Arduino Pro Mini with a battery by connecting the battery to the 5V pin, bypassing the regulator.

Somewhere else I read that the only difference between a 3.3V and 5V Pro Mini is the regulator.

Are these assumptions correct? If yes, can I use a 5V Pro Mini like a 3.3V by powering it with a 3V Battery on the 5V pin?
Pro Mini 5V has voltage regulator for 5V and 16MHz quartz.
Pro Mini 3.3V has voltage regulator for 3.3V and 12MHz quartz.

According to manufacturer specification Atmega328 chip powered by 3.3V cannot run 16MHz and needs at most about 13MHz. But it's official specification. Some chips are better than specification (faster) some are worse (slower). So you can try whether your Pro Mini is better or not. Or you can change quartz to 12MHz, if you can solder SMD parts.

Or better look for cheaper 3.3V clone, they are not much more expensive as 5V version...