The problem is that non-form elements are not tested against, right?
I haven't tested this approach but why don't you render an empty select-element that has the same name as the id of div and add a method that grabs the correct element:
HTML:
<form>
<!-- this is a dummy element -->
<select name="myElem">
<option>Testing form element</option>
</select>
<!-- this is my actual element -->
<div id="myElem">
Testing div
</div>
</form>
JS:
jQuery.validator.addMethod("myForm", function(value, element) {
var strId = $(element).attr('name');
var myActualElement = $('form').find('#' + strId);
var myActualContent = myActualElement.text();
return this.optional(element) || /regexwhateveryouwant/.test(myActualContent);
}, "Custom error message");
If you're not able to render a dummy select-element, you need to create it with jQuery.