Michael Lyons from Cork

My 3rd great grandfather Michael (Michl)Lyons was born abt 1835 in Cork. He appears to have emigrated to the US in Jan 1853 on the ship Anne. He was with his brother or perhaps cousin John born abt 1833.He came to Louisville Kentucky (my home town where our family still resides) immediately and was getting Post Offices notices as early as Sept 1853 as well as his brother John F Lyons. He was married to Bridget Finn here in Louisville in 1857. They first appear in the 1860 Census together Michael 25 Bridget 24 there daughter Mary 1 and John 27.

They named their children in this order:

Mary, John, Catherine, Michael, William, Winnifred

I commissioned a professional researcher years ago the baptismal records revealed two potential families that both had a Michael and John of this age. Micheal and Jane Buckley of Kilworth, and John and Catherine Murphy of Carragaline. Unfortunately these were all traceable to Australia. I've take DNA tests on Family Treeand Ancestry.combut haven't had any luck making a connection that would give clues as to the Parrish.
I also searched for their naturalization records but the possible matches here in the Louisville index were destroyed over time.

We just know from our family oral tradition that Michael was from Cork, but haven't had any luck pin pointing any further.

Thank you for any help!