Davidovicis. Proper names are best treated as slightly less mutable than other nouns when pluralizing. For example, for a family named Perry, one would write "the Perrys", not "the Perries".
The bind you're in is basically one of a poor match between spelling and pronunciation. If the name had been Anglicized as Davidovitch, we would pluralize it as Davidovitches and all would be well. Appending -es is basically trying to pluralize as if that were the case. Trying to apply a spoken pluralization to a written word when the written word already doesn't match its pronunciation is pretty much just piling breakage on breakage, though. Stick with pluralizing the written version based on its written form and the spoken version based on its spoken form.