Brightness when using 3.7v battery for powering 6 LEDs rated at 5v in a series
More details about your setup, including what chip you are using would be beneficial. But here are some things to consider.

I am assuming the specs in this datasheet are similar to lights you are using (same model, different manufacturer). The current ranges from 22.5 mA to 26.5 mA, which at 5 V results in 112.5 mW and 132.5 mW.

The first thing you can do to maximize the brightness is to use the designated power supply voltage for the LEDs. There is an internal voltage regulator that regulates the internal voltage to 4.3-4.7 V. It is important to note that these devices do not consist only of an LED light, but the chip that controls the color and uses SPI communication. A proper 5 V input is required to ensure proper communication.

I recommend using whichever setup allows you to provide 5 V to your system for the lowest volume. This Adafruit buck-boost converter will allow you to boost the 3.7 V from the battery you have now to 5V.

As for how long the battery will last, a quick search will lead you to the right equations and even some calculators already set up.

However, study up on the effects that current draw has on battery life. Draining large currents from a small battery will significantly reduce its life. Instead, whenever possible it is preferable to use a higher voltage battery and use a buck converter to reduce the voltage. This will allow a lower current to be drawn from the battery for the same power due to the higher voltage.