Update: The issue is solved
We noticed that our first two children had actually been born in my boyfriend's house, even though I was the host. We decided to try having children in my main farm house instead. Surprisingly, it worked and we were able to have two more kids in my house (so now we have four in total!).
Initially, the achievement still didn't pop up after the new baby arrived. However, instead of launching the save through the Co-op menu, I loaded it directly through Load Game. As soon as I entered the farm, the "Full House" achievement unlocked immediately.
Leaving this here in case it helps anyone else facing the same bug :)