Make sure to pack most of what you need for the afternoon including a diaper bag, snacks, local currency (euros) and your stroller.
The streets through the Gothic Quarter of Barcelona are mostly cobblestone, so although taking your stroller around isn’t exactly smooth sailing, it’s not too bad.
Anytime from 11 a.m. to 2 p.m. is great. Otherwise, 5 p.m. until 10 p.m. is also great. The timing in Spain is shifted back about three hours for what you might be used to for meals. For example, dinner is usually served at 9 p.m. rather than 6 p.m. Also, be mindful that most shops close between about 2 p.m. to 5 p.m. for siesta.
Spring and fall are the best times to visit Barcelona. The summertime is typically the busiest and most touristy and winters can be a little rainier.