As per our understanding on the scenario mentioned by you The page load rule is not being fired for "onload" & "DOM ready" condition . However the rule is being fired for "Top of Page", "Bottom of Page". We will suggest to debug your code by following the instructions in the below mentioned points and let us know the outcome of the above stated debugging options so that we can further deep dive into the issue.
1. Do you see any "asyc installation" error on the console . It is simply an alert that helps you recognize that the timing on the page is somewhat questionable and that DOMREADY has fired prior to all of the synchronous scripts loading as they should. If that happens, check to make sure that the pageBottom() callback is actually in the correct place immediately prior to the </body> tag. If not then these types of errors can occur. So please check whether the Footer code is placed just before the closing of </body> tag or not.
2. Try to manually configure the code in Customize page code and use an s.t() call at the end of code to check again whether the data is not being sent or not.
Have you tried using an event based rule on the click of the search with a link delay? That might give it enough time for the search to complete before the rule fires. The documentation is available here: adobe.ly/1MuDYdc