Should your heuristic for an A* search algorithm be the same scale as your actual weights?
I think you might have misunderstood the heuristic function.  It is supposed to give an underestimate on the distance from a node $n$ to the goal node $t$.  The closer this estimate is to the true value, the better A* will perform.
The only criterion for the heuristic function is that it never gives a higher value for $h(n)$ than the true cost of going from $n$ to $t$.
Let $\text{dist}(a, b)$ be the &quot;true&quot; distance from $a$ to $b$, and let $\text{dist}_0(a,b)$ be an underestimate (e.g. the Euclidean distance from $a$ to $b$ if you have a map).
If you are searching for a path from $s$ to $t$,
then $g(n)$ is defined to be $\text{dist}(s, n)$, whereas $h(n)$ is $\text{dist}_0(n, t)$.  So
$$ \text{dist}(s, n) + \text{dist}_0(n, t) = g(n) + h(n) = f(n).$$
There is nothing about scale of $g(n)$ and $h(n)$, except that $h(n)$ must give a lower value than (or equal to) the true distance from $n$ to $t$.