In order to match your first two desired characteristics, the material under load must 1) not be stressed beyond yield and 2) keep the resulting strain/deformation stay within the elastic range.
In your design, you need to first determine the beam geometry, arranged in a manner such that it is in balance and in equilibrium about the support joint C. Then determine the load P that will initiate the rotation but not overturn the structure. The last, find the resulting primary forces (M) due to P and weight W, and the secondary force due to the displacement (eccentricity).
The displacement $\delta$ is a function of P, E and the rotation angle $\theta$, and it relates to the linear strain $\epsilon$. By setting $\epsilon = \epsilon_y$, you can solve the elastic modulus E = fy/$\epsilon_y$ for the given load P. At this point, you can see it will take several iterations to reach the desirable elastic modulus.
One note to your design is that the beam is weakened at point B by the borehole, but it does not constitute a "hinge" until the combined stress has reached yield on the remaining sections (on sides of the hole), a plastic hinge has thus formed. However, you need to pay attention to the stress concentration and premature buckling though.
Another note is the repetitive/cyclical nature of the loading will leads to fatigue, which needs to be included in the stress calculations too.
The calculation is quite involved and tedious, therefore, it is recommended to seek technical help from a mechanical or structural engineer to select a material that is suitable for your application with the least weight.