The predicted Allied casualties for a mainland invasion of Japan were so high

Those estimates were made after the fact, in response to criticism. In reality, a mainland invasion was never in the cards at all. It's a myth. There's nothing about it in any of the letters or journals of the people making the decisions. There were two actual alternatives to the bomb:

1. Cooperating more with the Soviets. The Japanese refused to surrender in part because they were holding out a desperate hope that the USSR would intercede as a neutral third party in peace negotiations, when in fact they were just stalling for time while they redeployed their troops from Europe to Asia. The US and USSR had planned to issue a joint declaration calling for Japan to surrender at Potsdam, but Truman pulled out at the last minute when he heard that the bomb had been tested successfully. The soviet declaration of war was only days apart from the dropping of the bombs and the Japanese surrender.

2. Accepting conditional, rather than unconditional surrender. The Japanese had already offered to surrender on the sole condition that the emperor not be tried for war crimes. The US had every intention of doing that, and it's what they actually did after the war. However, Truman had promised "unconditional surrender" and he wanted the newspapers to call it that.

The decision was all about prestige and politics and not sharing the spotlight. It wasn't necessary.

This is a very long video about it but it's very informative and well sourced.