Question of selecting the right battery for 3V, 350mA Laser module.

As you've discovered, batteries marked 1.5V hardly ever give 1.5V (they give a variable value down to about 1V depending on load and lifetime)).

You need more batteries (higher voltage) and something like this: Pololu Step-Down Voltage Regulator D24V6ALV

The more batteries, the longer it will last (higher input voltage means less input current).