Well, since a 9V battery works, you can use two 18650's. They will most likely work, with such a huge capacity, you can keep it running for hours. If you can find a HOST (housing, typically a flashlight that has been gutted out and cleaned of stock components and ready to install your diode), you can put a laser in a flashlight. I prefer 18650's since they are nicely sized and have a large capacity. 14500's ... 10440's even AA's if you have enough.
But usually a Li-Ion battery works best as you can recharge. If you only want to use a single battery, use a boosting driver, which automatically boosts the voltage to meet the appropriate needed voltage. They're a tad expensive, but they are suitable for high current use, if you're gonna reuse drivers. But for now, stick to two 18650's or a single 9V battery and you can go lase as much as you want. Building a flashlight laser requires some degree of research on wiring, but its not that hard.
DIY: How to Make a High Powered Burning Blue/Violet Laser Pointer - YouTube
Your build is easily and clearly demonstrated here. However, a previous version of the diode, the 803T not 805T is used, and it comes from a laser sled housing. You can use two 18650's the driver and diode will be able to operate with no problem. Except for maybe some heat... but its no big deal if you have a short duty cycle. Used to prevent overheating of the diode.
duty cycle: time on = time off.
30 seconds on, then 30 seconds off...
Edit. If you use extra voltage, the largest chip on the drive board... the LM117 or 317, can't remember probs 117, will get warm. Excess voltage will be dissipated as heat. If you are using a high voltage source, BE CAREFUL and heatsink your thing.