You are very much misreading this. He isn't chasing dreamers to sign up OR spending an undue amount of time with that.
The Democrats want to save DACA as a reason to throw rocks come the Nov elections. They know it , he knows it, we know it. Right know they have the GUN debate keeping them busy, (in fact this may have been partial to kick the DACA can down the road).
Trump is documenting that he was ready willing and able to make a deal, so when they pull that shit up the road, he can prove it to be a prefabricated storyboard of no merit.
As an aside, by pressing them he is (realistically) pushing them to do nothing because they want to be obstructionists. This again, will backfire, because given a good turnout in November, they will loose their filibuster ability and will really have no control. At that point Trump/Repub majority will do what they like and that freakin' witch won't be able to do a damn thing about it.