Is it possible to power a 12v power board with a lithium battery, while keeping it charged via a solar panel. Or would I need to have two batteries and somehow automatically change them over when the battery was running low?
Without any sizing details, the question becomes "is the solar panel output greater than the input required for the board plus the charge required by the battery plus the losses" .
The solar panel output is dependant on sunshine hours and that can be estimated from data ygat can be found.
The load you should be able to work out from what your board consumes, what the battery needs and the losses - which, if the panel is close to the board and battery should be small anyway.
One battery should be sufficient anyway.