We know that $$P(F\mid G^c) = \frac{P(F \cap G^c)}{P(G^c)}$$
Thus, the denominator of our expression is just $1 - P(G)$
What is the numerator, however? I attempted to express the numerator using the exclusion principal for probabilities: ($F \cup G = F + G - (F \cap G)$), but, if I solve for the expression involving the intersection, this requires me to insert a union into the expression.
That numerator, $P(F\cap G^c)$, is as simple as you can get.
There are multiple other ways to write that, for example with a union inside a complement rather than an intersection outside a complement, but none of those other ways would be considered "simple". If you want to avoid complements, the easiest way is probably
$$P(F\cup G)-P(G)$$
or perhaps
$$P(F)-P(F\cap G)$$
If you look at a Venn diagram of two sets, $F\cap G^c$ is one of the "basic" regions of the diagram, not a combination of multiple other regions.
Is there some particular goal you have in mind for rewriting the numerator?