On second thoughts perhaps the idea wouldn’t work in countries that don’t have an effective lockdown. Could the infection wave of a less dangerous genotype overtake the number of infections of the more dangerous genotype before the population has been mostly infected already?

Yes if we act soon as we can spread the less dangerous version faster than the normal human-to-human spread of the dangerous version. For example if we were to send a postcard infected with the less dangerous version to 10 million people it would get out ahead of the dangerous version spreading normally.