Is it permissible to marry the son of zina whom his biological father acknowledged, given the scholarly disagreement regarding his lineage?

The matter is as you mentioned regarding the established difference of opinion concerning the lineage of a child born out of wedlock. If your cousin acknowledged him as his son, following those who permitted it, then he is his son, and he is not required to change his lineage. There is no harm in your marrying him, as he is not unlawful according to Islamic law, for Allah Almighty says: "And lawful for you are [all others] beyond these."