Avis and Budget are one and the same in Portugal. I've rented many times from Avis in Lisbon and the only negative is the long lines (similar to many other providers). Avis is on my + list for PT.

I would suggest checking 3rd-party sites like doyouspain, rental-cars.com, economycarrentals to see if you can find cheaper rentals.
Do watch the reviews - some providers are quite 'meh'.

Other providers to check - guerin, driveonholidays. Usually they are all cheaper through third-party companies.
I've also had good experiences with Europcar, Sixt, Hertz.

Avoid goldcar. Centauro can be 'meh'.