I too wouldn’t stay on Obidos - it’s worth a couple of hours. Caldas da Rainha or somewhere on the coast would be my choice of an overnight stop. I covered Alcobaca and Batalha in one day from Caldas. If your time is short, drop one. Fatima was my least favourite town in this area. I liked Tomar.
I think you are trying to cover far too much ground in your timescale. It will be a blur. Don’t underestimate how long it will take in Portugal to get around when you aren’t on the toll roads. I rarely do anything in the times that Google maps suggests - add a third plus time for photo and coffee stops.
Personally, I would drop the Algarve, as your time there is far too rushed. It’s a 3+ hour drive from Evora to Tavira on the rather boring toll roads. That leaves half a day to see Tavira, a pretty place. Tavira to Lagos is 2 + hours avoiding the toll road and going through more interesting places. It’s an hour and a quarter by toll road. The rocks around Lagos are interesting, but the folded rock formations stretch north up the western coast, so you could see the formations as a day trip from Lisbon, rather than the upheaval of moving accommodation again.
Nazare is my least favourite resort on this coast and IMO overrated by RS. It was more interesting a decade ago before it was overdeveloped and become a surfers hangout.
I seem to recall in a previous post that you are travelling this September. The Algarve is usually busy then, but this year in particular, it will be overrun with tourists. It’s one of the few places that the U.K. yesterday said is safe to travel to and the only beach destination, so apparently bookings have rocketed and prices are increasing rapidly.
Petrol and diesel is expensive in Portugal - more so than the U.K. and particularly if you are coming from North America. As your trip stands, sorry to say you will be spending as long in the car as you will actually visiting places. I haven’t visited Belmonte, but it’s a lot of driving just for one night there. I think your trip would be more enjoyable if you stayed in Lisbon - Caldas/coast - Coimbra - Douro - Porto - [Evora] - Lisbon, which is still a brisk pace.