It is not possible to be both a Buddhist and a Christian because Buddhism teaches all things without exception are 'anatta' ('not-self') & 'the elements' ('dhatu') as its core & fundamental principle; where as Christianity includes 'self' in the form of Christ & The Father as a core principle.
In other words, Buddhism is strictly 'impersonal' where as Christianity is 'personal', where salvation is based on a relationship between the 'small-self' and a personal god (Christ & The Father). For example, Luke 23:42 states:
Then he said, “Jesus, remember me when You come into Your kingdom!” And Jesus said to him, “Truly I tell you, today you will be with Me in Paradise.”…
Buddhism explains there are many types of liberation of mind (MN 43), which include liberation via loving-kindness & liberation via emptiness of self, the later being the foremost & unshakeable.
He keeps pervading above, below & all around, everywhere & in every respect the all-encompassing world with an awareness imbued with good will (loving-kindness): abundant, expansive, immeasurable, free from hostility, free from ill will... This is called the immeasurable mind-release.
And what is the emptiness mind-release? There is the case where a monk, having gone into the wilderness, to the root of a tree, or into an empty dwelling, considers this: 'This is empty of self or of anything pertaining to self.'
Now, to the extent that there is immeasurable mind-release, the unprovoked mind-release is declared the foremost. And this unprovoked mind-release is empty of passion, empty of aversion, empty of delusion.
Christianity is merely liberation via loving-kindness where as Buddhism is liberation via emptiness of self.
Also, Christianity teaches salvation is 'Eternal Life' where as Buddhism teaches salvation is 'dispassion' that occurs via the realisation of impermanence & not-self. In other words, Christianity is a doctrine of 'Eternalism' (the belief a self exists after death), which is a wrong view in Buddhism.