Yes. Suggest to the priest that singing "Happy Birthday" is inappropriate without a celebratory treat for the happy celebrants, and that the singing should therefore be moved to the parish social hall where coffee and doughnuts are served after Mass. Perhaps someone can stick a lighted candle in a doughnut for each celebrant to make the "family ritual" complete.
You may detect a slight edge to the answer above. You would be correct. The idea of singing "Happy Birthday to You" to one's fellow sinners in the presence of the re-presentation of Christ's sacrificial death on Calvary is, quite frankly, repulsive. It brings to mind the soldiers gambling over a dying man's clothes while the previous owner is dying a few feet away. Just because something seems like a good idea ("Hey! We're all family. Let's celebrate!" / "Hey! These are perfectly good clothes going to waste. Let's gamble!") does not make that idea appropriate in all contexts.
As for your idea of doing this after the final blessing, that is certainly an improvement over the current practice, but it is still inappropriate. The sanctuary is a place set apart for the worship of God. The social hall is the appropriate place for the celebrating the temporal milestones in the ordinary life of the parish.