It appears that the equilibrium is actually unstable if the truss doesn't have any thickness, so the center of mass is in the middle of the line segment AC. (call this point M)
Only if the center of mass of the truss is somewhat below M can the equilibrium be stable. The computation gets really hard if the center of mass isn't between A and C anymore, because this means that M need not be on the same vertical line as B.
If the center of mass is in the middle of the line segment at the point M, this means M will be exactly below B and a minimum of the potential energy of the truss is a maximum of m.
it isn't too hard to find BM as a function of x, if the lengths of AB and BC are l+x and l-x
(the length of the string is 2l). I hope you know the cosine rule.