According to the data sheet it is 125mA at the highest TX power so that would be only 0.46W with the 3.7V battery (assuming that calculation is correct, it’s been a while since I studied electronics..).

I was actually thinking of using one of the pre-built solar charging systems designed for cameras that have a USB interface. Not the cheapest solution and maybe a bigger panel than is really necessary but I’m guessing it may already have the circuitry built in so that it won’t output anything until it can output 5V.