Barcelona in Feb is roughly similar to Dublin in April ....
- can be cool ( 8-12 c) during the day and only 2-4c in late evening (colder up in Tibadabo or Montserrat)
- OR you can be lucky and it can be mild,almost balmy, some days with temps in late teens with pleasant sunny days.
It is so difficult to give an answer about weather nowadays- look outside! - but to be on the safe side , would you really want to risk a week in a city with two small kids when there is a good chance it could be wet, cool and windy ??
If do take that risk make sure that the Tv has good satellite selection - kiddies programmes in catalan may not appeal. That is, if you find a warm, well-located holiday apt in Feb.