Hello. Thank you for contacting the Toolset support.
There is no such feature available at the moment to search within the custom admin column you setup. This will require addition custom code that you will require to add that is beyond the scope of our support policy.
Thanks so much for sharing that link it sent me on the right path. I just noticed that for me that won't work because that is for custom post fields in the same Post Type, but for me it needs to search for the related custom post field. I digged a little further and found this:
This seems to be exactly what I am looking for, I am only wondering if you'd be able to take a quick glance and see if it will work for us because it was originally made for acf plugin, and I am not sure if using it with toolset will be an issue?