I have been a member of the Church for many decades and have never felt the Lord's Supper should be served before the invitational hymn. We want to wait for those who care to join our work, so we should serve it after the hymn. Do you agree?
The assumption is that most people decide to become Christians when the invitation song is offered. Biblically and in my experience people respond when they are taught and that can be at any hour of the day. No matter when the Lord's Supper is offered, there will always be some who decide to become a Christian after it has been served. There is no problem with missing the Lord's Supper because they were not yet a Christian at the time it was offered. The command is to wait for one another (I Corinthians 11:33). It is taken when the brethren gather together as a church (I Corinthians 11:18). If someone is not yet a part of the church, that command has not be violated.
Another way to look at this is to realize that when people knew the truth, they were baptized immediately. If someone wants to be a Christian, there is no need to wait until the invitation. "For He says: "In an acceptable time I have heard you, And in the day of salvation I have helped you." Behold, now is the accepted time; behold, now is the day of salvation" (II Corinthians 6:2).