It partly depends on your stamina (walking ... ) and desire to do the research about what you want to see (and wait in line to enter, if necessary), but I don't find it necessary to use a tour to see Sintra.
We did use a commercial provider for two days of "bus tours". Those get a bad rap here, but we got good value, with three or four widely separated places in one day, often not getting back downtown until 7PM. I don't mean to make decisions for you, but (unless you're a big Catholic), do you really need more "free time" at Fatima than the tour gave us? We were glad to get there with such ease, and no worries about public bus schedules, or where to find the stop. For comparison, the few days we had a car, we stopped at another Catholic pilgrimage site, Bom Jesu, and (except for the long staircase walking time ... a "feature" of the site) the private experience was not that superior.
I should add that we chose our bus tour provider partly based on the fact that the departure point was less than 1/4 mile from our downtown Lisbon hotel. But that's not absolutely necessary. The company was Portugal Tours, av. Sidonio Pais (in 1997!)