What, wait, no - if you want the properties to be the same in the ajax handler, you should submit them via ajax to the handler from the calling page. Otherwise people cannot dynamically customize those properties
Ok, I will try to explain the things better ...
In the main AjaxSearch snippet, you have a default configuration file (equivalent to the locked default properties set), you could have an additional configuration file (equivalent to a custom property set) and finally some properties initialised thru the snippet call. Until here, all is simple

But for the popup window launched thru an ajax request, you should transmit the user parameters (example &opacity=`0.5` or &parents=`128`). As AjaxSearch as lot of parameters, it is better to send only the "non default" parameters thru ajax. Not all. Like in this example:
<script type="text/javascript">
//<![CDATA[
asvar[asvar.length]='{"vsn":"1.9.0","adv":"oneword","sub":"","bxt":"Search here...","cfg":"&asId=`as2` &category=`articleCategory` &parents=`24` &moreResultsPage=`203` "}';
//]]>
</script>So, to send a limited number of parameters, I need to distinguish in the main snippet what are the default parameters (that will be retrieve in the second snippet (locked default properties set)) and what are the custom properties specifically added for an ajaxSearch on a page. This allow to have a unique hidden page with the second AjaxSearch snippet call for all the main ajaxSearch snippet calls. Whatever the custom properties added.
But I understand that the $scriptProperties array contains all the properties,
without the possibility to distinguish the origin. So, I can’t know which properties should be send thru javascript.