Let $(E,\mathcal E,\mu)$ be a probability space, $I$ be a finite nonempty set, $h:(E\times I)^2\to[0,\infty)$ be bounded and measurable and $$F(w):=\sum_{i\in I}\sum_{j\in I}\int\mu({\rm d}(x,y))h((x,i),(y,j))w_i(x)w_j(y)$$ for $w\in L^2(\mu)^I$.
How can we compute the Fréchet derivative of $F$?
I guess there are at least two options: Either consider the integrand as a function $(((x,i),(y,j)),w)\mapsto h((x,i),(y,j))w_i(x)w_j(y)$ or as a function $(((x,i),(y,j)),(a,b))\mapsto h((x,i),(y,j))a_ib_j$ evaluated at $(((x,i),(y,j)),w)\mapsto h((x,i),(y,j))w(x)w(y)$. It shouldn't matter at the end, but what's the "natural" (or easier) perspective?
First, note that finite sums are not a problem for Fréchet derivatives. Thus, I would recommend to only focus on one summand at first.
Then, try calculating the directional derivative at $w$ in some direction $v\in L^2(\mu)^I$. The result should give you a good candidate for the Gâteaux and Fréchet derivative.