This is a question inspired by and tangential to "A Question on 1, 2 ,3 Conjecture"—and certainly much easier!

Suppose one assigns a random edge weight among $\{1,2,3,\ldots,k\}$ to each
edge of $K_n$, and then sums the incident edge weights to each vertex
and assigns those as vertex weights.
For example, here is $K_5$ with edge weights from $\{1,2,3,4,5\}$:

(The top vertex has weight $13=3+1+4+5$.)
In this case, the vertex weights do not form a proper coloring, because there
are two vertices assigned weight $14$.

What is the probability that $K_n$ will be properly colored when the edge weights are chosen uniformly from among $\{1,2,3,\ldots,k\}$?

Empirically the function is well-behaved. Here is an accounting over $1000$ trials
on $K_5$ for each $k$ from $3$ to $20$:

As requested, a plot of Lucia's $e^{-\frac{\sqrt{\frac{3}{\pi }} n^{3/2}}{2 k}}$ for $n=5$ (with $k$ on the horizontal axis):