"Timing Issue" is the same as "the input box disappearing as soon as the search is performed"... they aren't separate suggestions... It's a "timing" issue because there isn't enough time to get the value before it disappears.
When you run _satellite.getVar('Search_Term') in your console, it executes the extraction... you didn't initiate the search, the input field is fully available to grab the value from the element....
When you are actually performing the search, the the search behaviour takes you from the page where you executed the search to the search results page... it's entirely possible that the input element is not reachable by the time the Data Element attempts to execute and get the value..
Without seeing the implementation or the behaviours this is the most likely scenario based on the fact that the Data Element works just fine when not actually submitting the form.
You could try to create a rules on "Blur" of the input element.. Holding the value to use on submit (rather than grabbing the value during submit), but if the user submits the form using their enter key, I don't know if this constitutes a "blur" action.
Or, you could ask your developers to try adding a short 100-200ms delay after submitting the form and leaving the page to load the results.. this should be small enough not to be noticed by users, but might just allow your original Data Element to succeed in getting the value.
I would work with your developers, show them the issue you are having. Work together in a Dev environment to try a few things like adding a slight delay to see if that fixes the problem...