Is it possible in NetSuite to filter lists on list/record type drop down line items to show only a set number of items based on a saved search? Meaning restrict some results in the list from being shown based on the criteria in the search?

If you are looking for filters for the items in the sublist of the transaction - use the transaction form and define the saved search in the top of the sublist fields. And if it is a custom field then define the saved search in the custom field's sourcing and default value tabs.

Essentially I have a custom multi-select field on the 'class' list, and it is pointed to the 'locations'. I set up a client script and a suitelet to search and return the values of the matching locations once a class is selected, but I feel like I'm stuck at that point because I can create an error message that defines the available locations and prevent the line from being validated until one of those is selected, but I can't actually manipulate the view of the list to only show the matching locations from the class record's 'location' selection setup that I created

you could have used dependent dropdown list instead of scripting. See the following suite answer https://suiteanswers.custhelp.com/app/answers/detail/a_id/102864

It sounds like I would need to create a custom list to do that though, right?

Or you can simply use the field's Sourcing and validation option