Does the board draw more than 1W sustained?
If it doesn't, a simple $0.10 reversed zener diode between it and the solar supply should work. Choose a zener with a breakdown voltage of about the minimum the board will start on. Maybe add an electrolytic or super capacitor in parallel to handle the start-up and transmit peak loads.
Then when the solar starts up in the morning it won't reach the board until there's enough juice to actually run it.
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.
https://www.bunnings.com.au/orion-smart-5w-ip44-outdoor-solar-panel-for-security-cameras_p0261685
