We have duties to ourselves, to others, and to God. God wants us to come to Mass, and the Church has declared that we have a Sunday obligation. Sometimes it’s appropriate to put our duty to others ahead of our duties to ourselves, but it’s never right to put either of those over our duty to God. If your ministry is making you late to Mass, you either need to start it after Mass, go to Mass on Saturday evening, or cease to participate in it.
It’s great that you’re engaged in ministry, but imagine showing up one day just to have those you minister to be delayed by 20 minutes every time. This is what the priest offering these TLM Masses must feel when you arrive late. They’re there to minister to you, but you’re always delayed.
Don’t let your ministry carve into your duty to God, which is to arrive to Mass on time.