Must an imam who is late for prayer due to a valid excuse pray with the delayed congregation, or is he permitted to pray at home for fear of being thought ill of?

A Muslim must strive to pray in congregation and join whatever remains of it, unless there are compelling circumstances. Fear of what people might say is not an excuse to abandon the congregation. One should be keen to pray in the mosque even if they are late for it, unless they know that the congregational prayer has ended, in which case they may pray it at home or with a second congregation in the mosque.