As stated by Ryan, using jQuery to do surround the control with a div is the cleanest way to go. It just requires a ContentEditorWebPart (CEWP) to be added to the site and the jQuery library uploaded to say a doc lib or, by using SharePoint Designer, any folder) include a reference to the jQuery library by adding a script like so to the page through the CEWP:
<script type="text/javascript" src="pathtojquerylibrary/jquery.js"> <script>
Then you would need to write some javascript code, also in the CEWP to select the control you want, add a div to the piece of html's parent you selected using jquery, then cut / paste the selected html into the new div.
it would look something like this (this code is not complete and certainly not tested):
$('query').parent().append('<div></div>').append($('query').html());
// where 'query' is the jquery selector for the control you want