Is there a way to bypass the jQuery error handler?

Forgive me if I'm interpreting this completely wrong, but seems you're looking specifically for the .ajaxError() handler, which you can use like this:

$(document).ajaxError(function(event, xmlHttp, options, error) {
  alert(error); 
});

Or you can bind it, it's the ajaxError event, just like a click event. This is only for AJAX errors instead of any jQuery throws, is that what you're after?

I just ran a test with the below. It will not throw an error if an HTTP 200 is passed back. The var 'result' contains whatever you expect back. Remove the datatype if you don't want to use JSON.

function ajaxRequest(request, url) {
  $.ajax({
    type: "POST",
    contentType: "application/json; charset=utf-8",
    url: url,
    data: data,
    dataType: "json",
    cache: false,
    async: false,
    success: function(result) {
      //this only happens on success
    },
    error: function(msg,XMLStatus,Err) {      
      ajaxError(msg,XMLStatus,Err); //Call generic error message
    }
  });
}

I tend to use this as a generic success/error method for web-service interaction.

/* Data must be prepared in a standard JSON format, either using $.toJSON() for objects, or stringbuilding for individual parameters */

/* General AJAX request handler */
/* serviceName is the full path of the service ie. "fullpath/services/service.asmx/method/" */

function ajaxRequest(data, serviceName, callbackFunction, async) {  
  $.ajax({
    type: "POST",
    contentType: "application/json; charset=utf-8",
    url: serviceName,
    data: data,
    dataType: "json",
    cache: false,
    async: async,
    success: function(response) {
      callbackFunction.Success(response);
    },
    error: function(msg,XMLStatus,Err) {      
      callbackFunction.Error(msg,XMLStatus,Err);
    }
  });
}

/* Sample use */
function getStuff(a,b,c){
  //Sample method signiture: getStuff(string a, string b, string c)
  var data = "{"
    data += "'a'" + ":'" + a + "'"
    data += ",'b'" + ":'" + b + "'"
    data += ",'c'" + ":'" + c + "'"
  data += "}";
  someParameterImayWantToUseInTheCallBack = "This was the test click button";
  serviceName = "/taccapps/samples/servicesample.asmx/getStuff/";
  ajaxRequest(data, serviceName, new sampleCallback(someParameterImayWantToUseInTheCallBack), true);
}

/* Sample callback */
function sampleCallback(someParameterImayWantToUseInTheCallBack){
  //someParameterImayWantToUseInTheCallBack is available to both the success and fail methods
  this.Success(response){
    //"response" represents the JSON response from the server. If it is a list/array it can be dotted into using the index.
    for(item in response){
      alert(response[item].a);
      alert(response[item].b);
      alert(response[item].c);
    }    
  };
  this.Error(msg,XMLStatus,err){
    //Standard HTTP error codes are found in the above
  };
}