QuantumCircuitOperator doesn't have options of its own. The "Diagram" property is calling the internal CircuitDraw function, you can get its exclusive options after filtering out some System options that it also takes:

FilterRules[
 Options[Wolfram`QuantumFramework`PackageScope`CircuitDraw], 
 Except[Join[Options[Graphics], Options[Style], Options[Rectangle]]]]

;

{"WireLabels" -> Automatic, "MeasurementWireLabel" -> "c", 
 "ShowWires" -> True, "ShowLabel" -> False, 
 "ShowMeasurementWire" -> True, "ShowEmptyWires" -> True, 
 "ShowOutline" -> False, "SubcircuitLevel" -> 1, 
 "GateOverlap" -> False, "MeasurementWirePosition" -> Top, 
 "HorizontalGapSize" -> 1, "ShowGateLabels" -> True, 
 "SubcircuitOptions" -> {}, "ShowExtraQudits" -> False, 
 "RotateGateLabel" -> Automatic, "GateLabels" -> {}, "Size" -> 0.75, 
 "VerticalGapSize" -> 1, "GateBackgroundStyle" -> Automatic, 
 "GateBoundaryStyle" -> Automatic, "GateShapeFunction" -> Automatic, 
 "ShowConnectors" -> False, "ShowGlobalPhase" -> True, 
 "WireStyle" -> Automatic, "ThickWire" -> False, 
 "DimensionWires" -> True, "IdentityGate" -> False, 
 "Arrowhead" -> 0.005, "Label" -> Automatic, "ShowGauge" -> True, 
 "LongOuterWires" -> True, "ShowWireEndpoints" -> False, 
 "ShowWireDimensions" -> False, 
 "OutlineStyle" -> 
  Directive[
   EdgeForm[
    Directive[Dashing[{Tiny, Tiny}], RGBColor[
     0.537254, 0.537254, 0.537254], Opacity[0.8]]], FaceForm[
RGBColor[0.898039, 0.898039, 0.898039, 0.3]]], 
 "BarrierStyle" -> Automatic}