This may or may not help you. If you try

FullSimplify[Collect[Eliminate[{
    x[1]==f11[x[2],y[2]],
    y[1]==f12[x[2],y[2]],
    x[2]==fx[x[1],y[1],x[3],y[3]],
    y[2]==fy[x[1],y[1],x[3],y[3]],
    x[3]==f31[x[2],y[2],0],
    y[3]==f32[x[2],y[2],0]},
    {x[2],y[2],x[3],y[3]}],
  {x[1],y[1]}]]

then it will quickly show you

expr1*x[1]+expr2*y[1]==0 &&
expr3*x[1]+expr4*y[1]==0 &&
expr5*x[1]+expr6*y[1]==0

Each of those six new expressions are large, but this has removed x[2],y[2],x[3],y[3] and perhaps you can see something in those expressions that will tell you something about the solution you are looking for.