Unfortunately it is not possible to search contents of 3rd party websites. If the demo site had the same content as the live site, then a possible solution would be to simply change the URL of the results with a filter function.
If that is your case, then this custom code might help you out:
Add this custom code to the functions.php in your theme/child theme directory (copy from line 3 only!). Before editing, please make sure to have a full site back-up just in case!Best,
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