NO dont drop an ice cub into a tank!!!!!!!!
you will shock your fish to death. frozen water in a chunk, it moves around spreading that cold all over the tank.
Bettas are a true tropical fish, where they come from their water temps range from low 70's to upper 80's depending on the depth of the puddle they are stuck in( yes stuck not choice), the monsoon rains elevate water levels and regulate the temps if the water and the fish move out and swim around the new streams and ponds that the rains create. They are territorial but thats out the window when in a bigger environment.
My betta is at a constant 79 degrees due to his tank mates needing a pinpoint temp setting of 79 degrees