Artchill – I wouldn’t try giving a gift to a police commander. It might make him/her suspicious as to what you were up too (and it’s probably illegal).
But for the others? Why not? You don’t have to mention the specific holiday. For that matter, there are other people besides Xtians around, and they might not like “Merry Xmas!” either. Give an appropriate tip, with the message, as mentioned above, “For all you’ve done for us this year.”
And if it’s local custom t tip, and you don’t, the “cheap Jew” label will surely be applied.