For a signal that is not corrupted with noise, how is the time-bandwidth product affecting the uncertainty or error introduced when extracting the fundamental frequency ? I mean, instead of choosing the fundamental frequency of the signal at $f_0$, you choose a harmonic $f_k$ at $k\cdot f_0$ and divide by $k$ to get $f_0$. How is the uncertainty related to $k$ for a given frequency resolution ?
If you a-priori assume that $k \cdot f$ is a stationary harmonic of periodic waveform of frequency $f$, that implies that you don't have any separable time locality of that harmonic. But any tighter bandwidth of a frequency estimation might come from the a-priori assumptions regarding the stationarity of the harmonics, not from the data.