The Catholic Church says yes, angels have free will. See for example paragraph 309 of the Catechism of the Catholic Church:
As purely spiritual creatures angels have intelligence and will: they
are personal and immortal creatures, surpassing in perfection all
visible creatures, as the splendor of their glory bears witness.
Paragraphs 391-392 of the CCC make it clear that the fallen angels made their own choice, freely:
Behind the disobedient choice of our first parents lurks a seductive
voice, opposed to God, which makes them fall into death out of
envy. Scripture and the Church's Tradition see in this being a
fallen angel, called "Satan" or the "devil". The Church teaches that
Satan was at first a good angel, made by God: "The devil and the other
demons were indeed created naturally good by God, but they became evil
by their own doing."
Scripture speaks of a sin of these angels. This "fall" consists in the
free choice of these created spirits, who radically and irrevocably
rejected God and his reign. We find a reflection of that rebellion in
the tempter's words to our first parents: "You will be like God." The
devil "has sinned from the beginning"; he is "a liar and the father of
Your question implies that you believe Michael, Gabriel, and Lucifer are each the "leaders" of 1/3 of the angels. That's not Catholic teaching. The theologians have speculated quite a lot about angels based on the clues in scripture, and I won't present it all here, but here are a few bullet points that may help you compare your theory to the Catholic understanding:
It is believed that there are nine "choirs" of angels ranked from the Seraphim (the highest), to the Angels (guardian angels like yours and mine).
Three angels are named in the Bible: Michael, Gabriel, and Rafael. (Rafael is named in the book of Tobit, which many Protestants don't read. It's believed he is the angel in John 5:1-4 but he is not named there.)
The fallen angels made their choice irrevocably at the moment of their creation, and cannot "change their minds". Likewise the good angels are not in any danger of turning to sin at some point in the future. Their choices are "locked in" like ours will be after we die.
There are many articles about Catholic teachings on angels, such as this one, which you can find by a web search.