Estimating a star's radius, temperature, and luminosity based on its mass
You could plot $L$ vs $T$ and compare your model result to the measured/estimated $L$ and $T$ of main sequence stars?  

You have not explained or labelled what your diagram is, but it isn't $L$ versus $T$; I suspect that it absolute $V$ magnitude versus $B-V$ colour.

The transformations between $L,T$ and $V$, $B-V$ are highly non-linear, especially at low temperatures (which is why your plot fails to reproduce the M-dwarfs) and involve folding a model atmosphere through some standard filter responses or using relationships between $T$ and $B-V$ and bolometric corrections and $T$. (e.g. Mamajek 2019)

EDIT: For the clarified question.

Your relationship for $T$ vs $B-V$ is not applicable at lower temperatures.

Your relationship between absolute magnitude and luminosity takes no account of the spectral energy distribution of the light from the star (i.e. it assumes the same bolometric correction for all temperatures, which isn't correct). You must calculate the bolometric magnitude from the luminosity and then apply a temperature-dependent bolometric correction to calculate a V-band magnitude.

The solution is to construct relationships that are valid over the full range of temperatures using the resource I referenced above.