FYI, you might find it more convenient to take a tram to Belem, not the train. The train station for Belem is not right in the center (Baixa/old town). It depends where you are staying. It is true that the trams were pretty mobbed in May (not just to Belem) and that the train may be less crowded and a little faster to Belem, but once you get to the train station and have made the connection, how much time have you saved?
Lisbon's train stations are poorly connected to each other. It would be nice if you could hop on a train at say Rossio and go to Belem - but you can't. You have to go to a completely different station.
I didn't do "zapping" with my Viva Viagem cards at all - I bought day passes most days in Lisbon for unlimited transport within Lisbon (including Belem) for the metro, trams, and buses, and I bought individual tickets for Sintra and Cascais. I had two different Viva Viagem cards as a result (different colors), one for Lisbon, one for the trains out. Buying day passes for unlimited use is just a personal preference of mine - I like to be able to hop on and off different trams and buses even for just a few stops and not worry about the cost of each ride.