What hard evidence is there that Charles Wesley was Arminian?

Actually there is quite a lot of evidence; start by reading his published journal for the years 1739-41, when he is in almost constant dialogue with Calvinistic Methodists like George Whitefield and John Cennick.
Then look at the two volumes of Anti-Calvinistic hymns which Charles published in 1740, and 1741 under the same title -- HYMNS ON GOD'S EVERLASTING LOVE.
He mounts a strong attack against Calvinism and an offers an Arminian soteriological alternative through out both books. This hymnals -- while they have been out of print for a long while -- are available in the reference set THE POETICAL WORKS OF JOHN AND CHARLES WESLEY, 14 vol. edited by George Osborn.
More recently, Horton Davies wrote a chapter on "Charles Wesley and the Calvinist Tradition," In CHARLES WESLEY: POET AND THEOLOGIAN, and I published a journal article on this "God's Everlasting Love: Charles Wesley and the Calvinistic Controversy," which was published in the EVANGELICAL JOURNAL. Basically, lots of information on this if you know where to look. JR Tyson

Yes, there are claims that both John and Charles Wesley were Arminians, but more often than not it is the theology of John that is under discussion. “And can it be” is a wonderful example of how Charles Wesley used hymns to present his theological views. The article in the first link below gives an interesting insight into the background and biblical basis behind the words.

Another article that sheds light on the theology of Charles Wesley is in the second link below: “Ye Need Not One Be Left Behind/For God Hath Bidden All Mankind”: Charles Wesley’s Response to the Doctrine of Limited Atonement by Charles Edward White, Spring Arbor University. Part of that article says:

“When John Wesley collected his brother’s hymns for the use of the people called Methodists, he opened the book with his brother’s birthday anthem, O For a Thousand Tongues to Sing. This song serves as an overture for the hymnal, introducing many of the characteristic themes of Methodist belief. Beginning with overwhelming gratitude and praise for Father and Son, it quickly moves to the proper human response of spreading God’s honor throughout the world. The intense personal experience of forgiveness, liberty, and cleansing comes next and then the declaration that all is of grace by faith fills out the first six verses. With verse six, however, Charles subtly moves from proclamation to argumentation. It is not by accident that against his Roman Catholic opponents he sings, “Look and be saved by grace alone/Be justified by faith.”

“With the introduction of the word “every” Charles arguably fires the first shot in a battle against Calvinism that will rage for the rest of his life. Along with the second hymn in the Collection and at least forty-three additional hymns in this volume, Charles wrote a thirty-six stanza poem, and published two volumes entitled “Hymns on God’s Everlasting Love” with forty-six more songs to combat what he calls “the poison of Calvin.”

“As we have seen, one of the topics which most exercised him was the concept that the atonement of Christ might benefit only a small portion of humanity. This idea he battled through arguments made from scripture, reason, and experience. Because his arguments are made in song, none of them can be fully developed or defended from all caviling. At least one, it must be admitted, depends upon a misunderstanding of the scriptural text. But when carefully examined, as we have done for two of the most-widely-known hymns on the topic, it becomes clear that the force of Wesley’s argument for the universal scope of the atonement is like the force of a landslide. While travelers may be able to dodge any individual rock, the sheer volume and weight of the whole phenomenon will overwhelm them.”

Part of an article (third link below) says this about Charles Wesley’s hymn “The Horrible Decree”:

“Charles Wesley was a prolific hymnist, with approximately 9000 hymns and sacred poems to his name. Among my favorites are “And Can It Be?” and “O, for a Thousand Tongues to Sing!” Welsey–along with his brother John–were also a theological polemicist, however, who wrote his polemics into his songs. One of his most blunt anti-Calvinist hymns is “The Horrible Decree,” which refers to the Calvinist doctrine of limited atonement.”

John Starke, pastor of preaching at Apostles Church in New York City and co-editor of ‘One God in Three Persons’ (Crossway, 2015), had this comment to make (fourth link below):

“For these reasons, it took me some time to come around to see the Wesleyan-Arminian theological perspective as something worth claiming. But I eventually did so. The sermons of John Wesley and the hymns of Charles Wesley were major factors for me. These are simply excellent, and gradually they drew me to the conviction that these Wesley brothers must have had a grasp of something important if they could keep producing things like that. The Wesleys teach a form of evangelical Protestantism that goes straight to the heart and changes lives. That's what drew me in to the Wesleyan way of thinking.”

A final thought from Jeff Robinson (PhD, The Southern Baptist Theological Seminary) who is a senior editor for The Gospel Coalition (fifth link below):

“The fact so many Calvinists love to sing the gospel-rich hymns of Charles Wesley, which John loved and printed and commended and sang, shows what Wesley meant when he said he was within a “hair’s breadth” of Calvinism.”

[1] https://www.thegospelcoalition.org/blogs/justin-taylor/charles-wesleys-and-can-it-be-background-and-scriptural-allusions/

[2] http://evangelicalarminians.org/wp-content/uploads/2015/05/White.-Charles-Wesleys-Response-to-the-Doctrine-of-Limited-Atonement.pdf

[3] https://georgepwood.com/2009/07/10/the-horrible-decree-by-charles-wesley/

[4] https://www.thegospelcoalition.org/article/youre-a-calvinist-right/

[5] https://www.thegospelcoalition.org/article/meet-a-reformed-arminian/