My question is, what would qualify as a "sacred function in honor of the Virgin"? Not Mass, although it's a feast day, right? Would a Rosary after Mass by the congregation apply?
The Feast of the Immaculate Conception is a Holy Day of Obligation. That means all catholics are expected to attend Mass. And what a joyous occasion!
The commemorations to Mary come from Scripture and are woven into that day's liturgy. For example,
Scripture
First Reading - Genesis 3: 9-15, 20
After Adam had eaten of the tree the Lord God called to him, "Where are you?" He asked. "I heard the sound of You in the garden," he replied. "I was afraid because I was naked, so I hid." "Who told you that you were naked?" He asked. "Have you been eating of the tree I forbade you to eat?" The man replied, "It was the woman You put with me; she gave me the fruit, and I ate it." Then the Lord God asked the woman, "What is this you have done?" The woman replied, "The serpent tempted me and I ate."
Then the Lord God said to the serpent, "Because you have done this, Be accursed beyond all cattle, and wild beasts. You shall crawl on your belly and eat dust every day of your life. I will make you enemies of each other: you and the woman, your offspring and her offspring. It will crush your head and you will strike its heel."
The man named his wife 'Eve' because she was the mother of all those who live.
Second Reading - Ephesians 1: 3-6,11-12
Blessed be God the Father of our Lord Jesus Christ, who has blessed us with all the spiritual blessings of heaven in Christ. Before the world was made, He chose us, chose us in Christ, to be holy and spotless, and to live through love in His presence, determining that we should become His adopted sons, through Jesus Christ for His own kind purposes, to make us praise the glory of His grace, His free gift to us in the Beloved. And it is in Him that we were claimed as God's own, chosen from the beginning, under the predetermined plan of the One who guides all things as He decides by His own will; chosen to be, for His greater glory, the people who would put their hopes in Christ before He came.
Gospel: Luke 1:26-38
In the sixth month the angel Gabriel was sent from God to a city of Galilee named Nazareth, to a virgin betrothed to a man whose name was Joseph, of the house of David; and the virgin's name was Mary. And he came to her and said, "Hail, full of grace, the Lord is with you!" But she was greatly troubled at the saying, and considered in her mind what sort of greeting this might be. And the angel said to her, "Do not be afraid, Mary, for you have found favor with God. And behold, you will conceive in your womb and bear a Son, and you shall call His name Jesus. He will be great, and will be called the Son of the Most High; and the Lord God will give to Him the throne of His father David; and He will reign over the house of Jacob for ever; and of His kingdom there will be no end.
And Mary said to the angel, "How can this be, since I have no husband?"
And the angel said to her, "The Holy Spirit will come upon you, and the power of the Most High will overshadow you; therefore the Child to be born will be called holy, the Son of God. And behold, your kinswoman Elizabeth in her old age has also conceived a son; and this is the sixth month with her who was called barren, for with God nothing will be impossible." And Mary said, "Behold, I am the handmaid of the Lord; let it be to me according to your word." And the angel departed from her. (RSV-CE translation)
It would also be appropriate to chant or pray this prayer.
My soul doth magnify the Lord,
And my spirit hath rejoiced in God my Savior.
Because he hath regarded the humility of
his handmaid; for behold from henceforth
all generations shall call me blessed.
Because he that is mighty hath done great
things to me, and holy is his name.
And his mercy is from generation unto
generations, to them that fear him.
He hath shewed might with his arm; he
hath scattered the proud in the conceit of their hearts.
He hath put down the mighty from their
seat, and hath exalted the humble.
He hath filled the hungry with good
things; and the rich he hath sent empty away.
He hath received Israel his servant, being
mindful of his mercy:
As he spoke to our fathers, to Abraham
and to his seed for ever.Hope this helps you understand the Scriptural basis for all feasts that acknowledge the Blessed Mother of our Savior, Jesus Christ.