I wanted to point out that, as you mention, with aggressive post-processing, you can routinely get mm vertical accuracy from GPS measurements. This has been done for several decades now, but it's not an off the shelf result. (Not a "cheap GPS device")

The post processing technology was pioneered by NASA among others. In the western US there is a network of over 1000 such monitors, anchored very deeply, that use these methods. Here is the time series from one in the Bay Area: http://www.unavco.org/instrumentation/networks/status/pbo/ov...

In that plot, you can see lots of interesting effects. First, the virtually continuous lat/lon velocity. Second, the annual vertical trend which usually has to do with groundwater. Additionally, smaller and larger perturbations from all kinds of sources from nearby construction to large scale seismic effects like distant earthquakes.

GPS of this kind is a complementary technology to the fantastic InSAR measurements featured in TFA. Both are widely used in geodesy to measure seismic deformations and deformations due to groundwater extraction. The two approaches offer different temporal, spatial, and accuracy trade offs. For a building like this, if you really wanted to measure subsidence, GPS would yield more temporal information than InSAR.

If I recall correctly, the fundamental limit to the GPS accuracy is uncertainty on tropospheric propagation of the radio signal.

I don't know if this technology is readily available for one-off commercial applications, but it wouldn't be surprising. The required accuracy is clearly within reach, and the software to do the post processing is widely licensed (https://gipsy-oasis.jpl.nasa.gov).