I am interested in what the data does say, but we should articulate what this discussion is about. I take your post to mean that:

  - access to affordable childcare increases fertility rates
  - inflows of lower-skilled workers to an area would increase access to affordable childcare
  - a way to increase the quantity of lower-skilled workers would be through immigration

The 3rd point above I think is trivial to accept, I am inclined to believe the other two points but would want to see the evidence that this is actually how things shake out.