Did you consider taking the Malev flight from Budapest to Tirana, and hiring a car from there?
I would allow about 5 hours from the border to Tirana and a further 3 from Tirana to Vlore. You should be able to complete the journey with ease in those timescales.
There's a new tunnel opened recently on the Kukes - Durres stretch which has reduced the journey by about an hour. Although the roads have improved greatly over the last couple of years, do not expect motorway driving conditions, perticularly in Northern Albania where the roads can be a little primitive. The roads between Tirana - Durres - Fier are pretty good these days, however the final section between Fier and Vlore can be a little 'rustic' in places.