Have you considered calling the Avis rez line and asking for help, and for what the best rates available are?

Have you checked into renting in Lisbon metro area, and not at the airport? Once in a while, this yields lower rates. Are your dates firm? Have you checked into making your pick up either the day before or the day after? Have you tried splitting the rental into two parts? Sometimes that will give you a lower rate for part of the time, instead of giving you the higher rate for the whole time.

I have never rented in Portugal, so I can't swear by any of these AWD codes, and some of them might not work in Europe at all, but it can't hurt to try, right? Check out:
G298100
K851000 (Hhonors)
A153700
B149870
Z652297
L358190 (BMW Car Club of America)
D833357 (American Society of Travel Agents)
A074800 (Citicorp Leisure Travel)
B908200 (Jackson Travel)
H854900 (Avis Special Weekend Rate through 06/30/2011)
G417900 (Kellsport/Kellwood Travel)

Good luck!