Change the monotonicity of a function but preserve its convexity

Question

Consider the function $f : \mathbb R^2 \to \mathbb R$ defined as $$ f(x) = \exp(x) + \exp(y) $$ $f$ is convex in $(x,y)$ and it is strictly monotone increasing in $(x,y)$ (in the sense that, if $x_1 < x_2$ and $y_1 < y_2$, then $f(x_1,y_1) < f(x_2,y_2)$). Does there exist a function $g$ such that

1. $g$ is invertible.
2. The composition $g \circ f : \mathbb R^2 \to \mathbb R$ is convex and strictly monotone decreasing in $(x,y)$.
3. $g(1) = 1$.

If condition 3 is too strict, I'm willing to drop it.