You don't need a ground rod.
You can do that with the inverter on one battery bank with the shore power cord plugged into the inverter and the converter to charge the other battery bank. You can't have the inverter and converter on the same bank (loop-drains battery)
You can have the two solar sets each with its own controller. One can be on each battery bank. Or you can put both controllers in parallel on the same battery bank, or you can combine the banks and have all the solar on that (but now the inverter can't run the converter, but it might not need to if the solar can do it all)
What's missing as a way to run the converter to charge the batteries when there isn't enough solar for too long. Usually a generator does that.
