That's exactly what we do: submitting a search/filter form through a POST request which creates a record in the database representing the search/filter criteria and then redirects to the same endpoint with the ID of that record. The endpoint looks for the ID in the query, attempts to load the same record, and if found, executes and returns the search results.

It works very well, is completely transparent to the end user, has automatic query logging, and provides a clean URL that can be easily shared.