The first step is to find the x and y coordinates of the corners for a given z. And to do that you need to work out a formula for the cone. Here' what I would do: Looking just at the first octant, a cross section at the x,z plane gives a line from x= 0, z= 1 to x= 2, z= 0. That is given by z= 1- x/2 or x= 1- 2z. Similarly, a cross section at the y,z plane gives a line from y= 0, z= 1 to y= 1, z= 0. That is given by z= 1- y or y= 1- z.
So at a given "z", the cross section is an ellipse with semi-axes 1- 2z and 1- z. The equation of that ellipse is $\displaystyle \frac{x^2}{(1-2z)^2}+ \frac{y^2}{(1- z)^2}= 1$.
Now we can say that we want to maximize the volume, V= xyz, subject to the constraint $\displaystyle A= \frac{x^2}{(1-2z)^2}+ \frac{y^2}{(1- z)^2}= 1$
To solve that using "Lagrange multipliers" set $\displaystyle \nabla V= \lambda\nabla A$.
That, together with the constraint, give four equations for x, y, z, and $\displaystyle \lambda$.
at z = 0 , your equation gives a cross section
$\displaystyle x^2 + y^2 = 1$
which is a circle. It should be an ellipse.
The cross section should be
$\displaystyle x^2+4y^2=4(z-1)^2$
and the volume = 4xyz