While I can not answer your specific question, I would suggest that you check into car rentals arranged from the US. I know this sounds counter-intuitive, but I've been told by many folks that it is cheaper to arrange for car rental from Canada. My last trip I did so and the rates were very reasonable. We can't post links to ads on this forum, but if you would like me to send you a link to a Canadian site (as an example) send me a PM and I will send you the link.
1, HIRE for a long time. This is fine in the winter - we did a deal with a local firm and paid about 240 euros per month for a 3 month old Fiesta. It was great, and there were no servicing costs etc. but the problem is that from April to September the rental firms won't be remotely interested in a long term hire as they can rent them to short term tourists for 300 per WEEK.
2. LEASE - yes there are leasing companies in Portugal same as the UK, but typically car leases here are over 3 years . they are shockingly expensive as well - when we were looking, a base level golf was around 500 euros per month with the IVA added on. Cars are very expensive to buy here.
The other thing with leasing is that it is seen like any other form of credit, so you need a boatload of paperwork - residency, tax return, wage slips etc.
We eventually managed to get finance (HP) to buy our car over 5 years. We had to pay a deposit and the paperwork took hours. We got finance direct from a garage - the bank wouldn't touch us, despite the fact we trasfer a good couple of thousand Euros into Portugal every month.