J.J. wrote:
"Is it wise to place a lightning rod (or rods) right under the
antenna---?"

Yes. It is a good practice to help nature to neutralize its excess
charge in the shortest time and distance. The sooner this gigantic force
is diluted, the safer everything and everyone is. But, if your outside
plant is impervious to damage, you may choose to put your arc gaps and
ground rods back at the station ends of your transmission lines.

Best regards, Richard Harrison, KB5WZI