Two quick comments - well, it morphed into more.
The extra energy in the spring can ultimately be considered to be due to the electromagnetic fields that hold it together. Electromagnetic fields have energy, and you need to include it to get the total energy. I'm not aware of any good simple model of the electromagnetic field in a spring. If you try to consider the energy of an electromagnetic field in a point charge, for instance, you get infinity. So it's not clear exactly how to model it, though it is clear that's where the energy is stored.
The second comment is that most of the above is a special relativity analysis, and the conclusions are about the special relativistic mass. It turns out there are at least three sorts of mass in GR, so the situation in GR is more complicated. If mass were as simple as it was presented, GR wouldn't need three different formulations of it.
In particular, if you have a stationary metric (roughly speaking - one that isn't time-varying), pressure in GR causes "extra" gravity, above and beyond the special relativistic mass increase. This extra effect of pressure reflected in the defintion of the Komar mass density, which is rho + 3P / c^2 in reasonably flat space-time. (THere are some other factors needed if you have significant gravitational time dilation anywhere).
Defininig the Komar mass requires that you have a stationary metric - more exactly a timelike Killing vector. Without this property, the above formula isn't correct and the Komar mass isn't defined.
IT's quite common to have a system that's either stationary or quasi-stationary. It's not quite clear how to compute the error when the system is quasi-stationary though, at least i've never seen an analysis.