Cheapest Method for Lip Augmentation?

It would of course be difficult to alter the lip shape without expending funds for even a filler.
However, it appears the maxilla is long and might benefit from a LeForte I procedure. You may wish to consider consulting with maxillofacial or craniofacial surgeons to see if there is a possibility your insurance might cover the procedure.

This is my philosophy when it comes to these sorts of things. There is a correct way to do things and the incorrect way. To fix a problem, there is a solution and that is the way to go. Anything else will result in expense with no results.

Unfortunately, there is no "cheap" way to augment the lip. You may want to go with fillers temporarily, to see how you like it.  Other more permanent fillers like alloderm or even fat injections may be longer lasting.