What is the QFT prediction for the electron $g$-factor to high precision?

The experimental results are indeed substantially more precise than the theoretical calculations. The precision of the theoretical result is limited by a couple of things. Firstly, in order to compare a prediction to the experimental number, another experimental input is needed to get the value of the fine structure constant. There are other precise measurements that may be made in QED, but comparisons are ultimately limited to the (experimental and theoretical) accuracy of the second-best-known quantity. For this reason, the measurements of the anomalous magnetic moment is often described not as a verification of the theory but rather as—assuming the theory to be correct—the most precise measurement of the fine structure constant $alpha$.

The other limitation comes from that fact that if you want to go to really high precision in calculations of $g_{e}$, you have to go beyond just QED. For pure QED calculations, going to higher an higher orders may be tedious and involve a lot of integrals, but how to do the calculation is known. In fact, modern calculations at high orders in $alpha$ are typically partially automated; computers are programmed to do the algebra to reduce the results from a large number of Feynman diagrams to a human-tractable calculation. However, to go to really high precision, it is necessary to include additional interactions beyond electromagnetism. For weak ($W$ and $Z$) interactions, this is not too challenging, but including strong interaction effects is really hard. For the muon anomalous magnetic moment $g_{mu}$, the error has been dominated by effects with virtual hadrons for decades; such terms are of lesser relative importance for the electron calculation, but they are present and will make a key contribution to the theoretical error.