5x ws2812 could draw around 300mA, if all 5 LEDs are at maximum brightness white.
A 9V battery has a high internal resistance, which means it can only give a few tens of mA before it's voltage begins to drop. The convertor board would help, reducing the current requirement to perhaps 160-170mA at 9V, but that is still probably more than what the 9V battery can give.
But if your project has room for that convertor board, it has room for a larger and better battery, which is a far better solution.
For the smallest suitable battery, you could look at a Li-ion or Li-Po battery. But these batteries need to be protected from over-discharge, or they can become dangerous. Small protection circuits can be purchased cheaply on eBay/AliExpress. A special charger would also be needed, of course. These batteries have a nominal voltage of around 3.6-3.7V. That's enough to drive the ws2812 until the battery is more than half drained. At that point, the blue LEDs go first, then the green. The Pro Mini will probably continue to work down to those voltages also, but if it doesn't, you could swap it for an 8MHz, 3.3V Pro Mini.