PDC is busier, brasher and a more touristy resort than PB so it really depends on what you want. I've been to PB most months of the year over the last 20+years and have never found it particularly quiet after 9pm. There is always people around and bars and restaurants open where you can sit and people watch at that time. After 11pm, yes it does go quiet, but it tends to do that in the summer as well, with the exception of the 2 commercial centres. It really depends on how late you're going to be out with a young child.
PB does have it's own micro climate and tends to be a degree or two warmer than the rest of the island, but you would need to ensure the pool is heated as the sea will be warmer at that time of the year, depending on the weather.
If you want a private villa with pool close to town in PB, look on Los Calamares. You'll not get much closer. 5 mins walk to the promenade. We stay on there every visit. All the restaurants on the island I've ever been in have always been child friendly. The Spanish adore kids. There is a great place in PB called Kikoland, which is attached to the Princesa Yaiza hotel, but is open to the public for a fee. They have all sorts of activities for children of all ages. There's also a kids playground on the main road, next to the church.
The bus timetable between PB and PDC is in this link.
http://tinyurl.com/pn4vu9o