Buddhism is incompatible with Christianity.
Buddhism rejects the concept of a permanent self, or an immortal soul.
In Christianity, the concept of "self" and the immortal soul are pervasively important. In Christianity, it is "self" that sins (Ezekiel 18:4), is resurrected (John 5:29, Romans 8:11), and is eternally judged (Matthew 7:23, Revelation 14:11). It is "self" that is offered eternal forgiveness through Jesus Christ (John 3:16).
Indeed, Jesus the Christ is a permanent "self", and always has been (John 1, Colossians 3:1, 1 Corinthians 15).
Without self, the Christian doctrines of sin, resurrection, judgment, and forgiveness become utterly meaningless. Jesus Himself ceases to be an immortal individual, the only begotten Son of God.
Thus, to embrace Buddhism is to reject the concept of "self", and to reject the concept of "self" is to utterly reject Christianity.
On the flip side, to accept Christianity is to recognize the reality of "self", the reality of sin (Romans 3:23), and the reality of our need for forgiveness, which is only available through the immortal Jesus Christ (Romans 6:23, 10:9-10).