Modelling the planetary heat transport of small bodies in the early Solar System allows us to understand the geologic context of meteorite samples. Conductive cooling in planetesimals is controlled by thermal conductivity and volumetric heat capacity, which are functions of temperature ($T$). We investigate if the incorporation of the $T$-dependence of thermal properties and the introduction of a non-linear term to the heat equation could result in different interpretations of the origin of different classes of meteorites. We have developed a finite difference code to perform numerical models of a conductively cooling planetesimal with $T$-dependent properties and find that including $T$-dependence produces considerable differences in thermal history, and in turn the estimated timing and depth of meteorite genesis. We interrogate the effects of varying the input parameters to this model and explore the non-linear $T$-dependence of conductivity with simple linear functions before applying non-monotonic functions for conductivity and volumetric heat capacity fitted to published experimental data. For a representative calculation of a 250 km radius pallasite parent body, $T$-dependent properties delay the onset of core crystallisation and dynamo activity by $\sim$40 Myr, approximately equivalent to increasing the planetary radius by 10\%, and extends core crystallisation by $\sim$3 Myr. This affects the range of planetesimal radii and core sizes for the pallasite parent body that are compatible with paleomagnetic evidence. This approach can also be used to model the $T$-evolution of other differentiated minor planets and primitive meteorite parent bodies and constrain the formation of associated meteorite samples.