Powering a 5V Arduino Pro Mini with 3.3V
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...