Hi,
Thank you for the details, it helps us a lot!
To show custom field contents within the live search content fields, you can use the advanced title and content fields feature. That should take care of that.
The results page is a different story, the plugin can not change the texts there. You may need to add a different hook for that. I believe the results page usually shows the excerpt by default, instead of the content. Maybe applying your custom code to the the_excerpt hook may simply do the trick.