I've often shared my story of having been raised in a tradition that focused on an individual, personal, sincere effort to get saved that led me to a very dark place spiritually. My struggle was whether I had "meant it" when I "asked Jesus into my heart"; and so found myself in a state of constant fear and doubt begging God to save me. And the harder I tried, the more I wanted to be spiritual, faithful, sincere, the further and further away God seemed to be. Jesus grew remote, distant, and angry in my mind.
It was the work, primarily of Lutherans, but also Reformed folk, to speak of the pureness of grace that stilled the wild storms of fear and doubt in my soul. It was this experience which ultimately brought me into the Lutheran tradition, where I remain. But at that early period of my life I was, arguably more inclined to the Reformed/Calvinist side. The Lutheran tradition with its very "Catholic" way, vs the much more familiar "Protestant" way of the Reformed tradition was closer to where I was as a born-and-bred Evangelical Protestant (mixed non-denominational, Baptist, and Pentecostal upbringing). Ultimately, however, it was the Lutheran focus on Grace rather than the Reformed focus on Divine Sovereignty that became the central issue: The Calvinist schema is one that is highly logical, rational, beginning with certain axioms rooted in Scripture and then works up from those axioms toward theological conclusions--it's all very self-consistent and rational. Lutheranism, on the other hand, just isn't interested in being rational; and yet where Lutheranism says that yes God has chosen us, because that's what Scripture says; Calvinism says that yes God has chosen us, but that must by necessity mean that God has passed over, neglected, or otherwise chosen by negation the rest who are damned and irreparably reprobate. Yet in Scripture I could find nothing about God picking and choosing who would and would not be saved; but rather the words of comfort: God has chosen you; and at the same time Scripture time and again emphasized that God's love is for all, that Christ died for everyone; that there is a universal mercy in the Gospel, and that Christ's work is universal in scope (not that all will be saved, but that God wills that all be saved).
It is because the word of God declares who God is, as revealed in Christ, that I ultimately could not follow the Calvinist path. I could never shake the sense that Calvinism obscures God behind a veil of Sovereignty where the light of the Gospel cannot fully shine. A theological mist that obscures the pure visage of Christ the God-Man and Suffering Mediator.
I don't know if this is the kind of argument against Calvinism the OP would be interested in. I think there are plenty of raw biblical arguments that exist, but I wanted to present something more personable. Less a "Why you shouldn't be a Calvinist" and more of a "Why I'm not"
-CryptoLutheran