this is something of a stretch, but my great-grandparents' surname was Meshurisman, they came from Balta, and had connections in Odessa and Kishinev. My grandfather changed the surname to Mishuris, and I've seen the original surname spelled Meschurisman, Mishurisman, and many other variations.
Could Meschmar be a shortened form of Meschurisman?
I believe the original meaning of Meshurisman is something like "servant", perhaps someone's occupation.
Oakland, CA, USA
Mishuris, Toberman, Polisar, Flickstein from Balta Ukraine;
Lewak, Goldberg, Kier, Flaksman, Dwinsky from Drohiczyn, Kobrin and Chomsk in Belarus.
Winzelberg, Wertheim(er), Silberstein from Czchow, Poland;
Lieberman, Finkelstein from Balbieriskis, Lithuania;