are there any stable four membered chelate rings?
Four-member rings can be formed with simple oxy-anion ligands such as nitrate.  An example with iron as the central ion is given here.

Reference


G. R. A Wyllie, O. Q. Munro, C. E. Schulz, W. R. Scheidt,
"Structural and Physical Characterization of (Nitrato)iron(III)
Porphyrinates [Fe(por)(NO3)] — Variable Coordination of Nitrate",
Polyhedron, Oct 10 2007, 26(16):  4664-4672.