teaching - Undergraduate Derivation of Fundamental Solution to Heat Equation

I think the term fundamental solution (at least sometimes) conventionally includes the integral around your $K$. I will assume this. If I recall correctly then the following argument is from "Partial Differential Equations" by Strauss.

A particularly simple solution follows from the self-similarity principle, i.e.

If $u(x,t)$ is a solution then so is $u(cx, a c^2t)$

This suggests looking for a particular solution of the form $K(x,t) = g(p)$, where $p = frac{x}{sqrt{4at}}$

Substituting $g$ into the heat equation leads to the differential equation

$$g''+frac{p}{2}g' = 0 $$

Then the fundamental solution as above follows from solving this.