Is this paper attributing a change in polar wander to climate change?
The term ‘polar wander’ is ambiguous. In this case it's got nothing to do with the magnetic poles, but rather with the Earth's spin axis. If you think about a desk globe mounted on an axle, the poles under consideration here are the points where the axle intersects the globe's surface.

The paper is essentially saying that climate changes are changing the distribution of ice and water on the surface of the planet, and this in turn is shifting the spin axis slightly. You're right that the mass of material on the Earth's surface is dwarfed by the mass under the surface; however, if you look at the paper you'll see that they're talking about miniscule shifts in the axis. The changes shown in Figure 1 are on the order of tens of mas (milliseconds of arc) over a few decades. One mas is one 3.6-millionth of a degree, or about 3.09 cm as measured on the Earth's surface. So, at least at first glance, the changes don't seem too implausible.

You mention isostatic response as another concern, but this is a slow process which doesn't really come into play on the decadal time scales considered in the paper. Fennoscandia and northern North America are still rebounding isostatically from the last ice age.