I don't *think* that the SHB will increase because of sugar water. The trick is to make sure that it doesn't give them a place to hide or lay eggs. I think that the larvae need a protein or lipid (wax?brood remnants?) source to grow, but don't quote me on that....
It isn't the actual beetle that is the problem eating honey/syrup, it is the larvae. I don't think that a feeder would give them a place to have larvae, unless you are disturbing the hive to put it on.
And no grease of pollen patties, this is an open invitation...
I've been watching them in my obs. hive, its been interesting. There is one place b/w a frame and the glass that the bees can't get to so they corral the beetles there, up to 7 of them. There is open hostility to the beetles, lots of pinching, but the bees can't do much to the smooth curved shells, not when the beetles "cower" so no appendages are sticking out. And boy are they fast!!!
What I did do was to stuff some pollen sub through the feeder mesh, since that is the only place to feed them. This gave the beetles a place to lay eggs, and a place for the larvae to grow. There was an infestation, which I think I helped by stirring it up and giving the bees access to it all. It was interesting watching the beetles slip through the #8 hardware cloth to escape (and get crunched, ha ha 8-).
Rick