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.

The geographic north pole, which is the Earth's spin axis, has seen significant shift since the end of the last ice age, mainly due to melting of the North American ice cap. As N. America got 'lighter' the geographic north pole moved towards Hudson Bay. Now, with climate change Greenland is starting to melt so that the polar shift has changed direction - towards the UK in fact, but the total measured effect over the last few decades is only a few tens of metres.