I do not know whether any of the places you mention require that you obtain a timed-entry ticket for the free day. I wouldn't be surprised if the Picasso Museum does. I imagine Rick covers this in his book.
To a Spaniard, lunchtime more or less starts at 2 PM, so I'd expect some folks to exit the museums around 1:30 or so. Whether that will help at the Picasso, I do not know. It was a nightmare late in the afternoon on a regular (not free) day in August 2016. I'll say the same thing to you I say to others: Skip that museum if you don't have a serious interest in it. Barcelona has many other art museums, and only the Picasso tends to be packed shoulder to shoulder so that it is difficult to get close enough to the art to read the descriptive material posted beside it. Perhaps if you were to arrive an hour before opening time on the free day...
I actually went to the MNAC on a free Sunday that same August, and there was no significant line to contend with. It's large, but not as large as the Prado. There were lots of folks inside the museum, but not to the extent that you couldn't get up close to the art and move around at will. Don't miss the church frescos and the modernism collection.
You might get lucky at the Palau Guell, which I think also has some free time on the first Monday of the month. I don't claim that it is a substitute for Casa Mila/Casa Batllo, but it's worth visiting, and as expensive as everything is in Barcelona, free is wonderful.
I went to the Prado on Museum Day in May 2016, but it was a Wednesday, and it seemed apparent that the tourists didn't realize the museum was free and the locals, naturally, were at work during the day. There was no line to enter. I guess it's like the Louvre: Everyone wants to see a few famous paintings. On the day I was there, there were clots of people standing in front of Las Meninas and the Garden of Earthly Delights. Otherwise, there was nothing to lessen my enjoyment of the museum.
Half the people at the Reina Sofia (not on Museum Day) seemed to be standing in front of Guernica. I have no memory of any sort of crowding at the Thyssen.
The Picasso Museum is a whole different matter.