stellar structure - How well can we in principle determine $T_{textrm{eff}}$ of a star?

This is a question about the basics of astronomy, which I have never happened to see a good discussion for. It is about how well would we be able to measure effective temperature of a star, if we had any arbitrarily perfect measurement devices.

Here is some context. Canonical definition of $T_{textrm{eff}}$ of a star is based on its bolometric luminosity $L$ (total electromagnetic energy radiated by the star per unit time) and its photospheric radius R (radius, at which the optical depth at a given wavelength is equal to unity). This way, the definition specifies $T_{textrm{eff}}$ through $L=4pi sigma R^2 T_{textrm{eff}}^4$, where $sigma$ is Stefan-Boltzmann constant.

The definition clearly alludes to black-body law. Many stars, including our own Sun, have a spectrum that does not follow it. For this reason, one often talks about another effective temperature, which is the temperature of stellar material at photospheric radius, and which can be determined by examining stellar spectrum. There are a few more complications to that, but let's put them aside.

Determining $T_{textrm{eff}}$ is extremely important in characterising stars, therefore there exists a variety of methods of measuring it, and naturally researchers strive for obtaining the best possible precision.

Hence, the question: How well can one in principle measure $T_{textrm{eff}}$, if one could have arbitrarily perfect instruments?

Edit: I would like to see a quantitative estimate in your answer. Is the best possible precision for $T_{textrm{eff}}$ of order $10textrm{K}$, or is it $1textrm{K}$, or some $10^{-4}textrm{K}$, or can we measure it arbitrarily well?

Here are just a few sources of uncertainty/arbitrariness: convection in stars, dependence of photospheric radius on wavelength, limb darkening, stellar variability, to name a few.

I would encourage the answers to be in the format "Source of uncertainty" - "Simple derivation" - "Estimate of the effect". If there are more than a few estimates, I will add a summary of them in the question or in a separate reply. Please, also feel free to edit the question if you might like to.