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Although your panel array is capable of supplying 80 amps. It is the battery charge rate that the relay needs to handle. Your panel array will not be putting out the entire 80 amps into the batteries. Typically, float charging a good battery array only requires a few amps. The only time high amperage would be needed is if the battery array is almost discharged completely.
The LED strip could be placed around the base piece rather than under the top piece. This would allow a mains connection. You would then need to determine how to route the wires across/through the table however. Also using a lower voltage source of 2.5-3 volts would dim the LED's down a bit such as a previous poster desired. Your project is a great idea and inspires me to do something similar with a slice of black walnut I have from a recent logging job at my property.