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.

Solutions are such that the coefficients are not independent of each other.