It's a start. But the real battle will be in the Senate.
The White House has statements of support for both H.R. 3003 – No Sanctuary for Criminals Act and H.R. 3004 – Kate’s Law on its site indicating that President Trump would sign them. The challenge though would be in the Senate.
The Senate is the last best hope of the left. The left is gambling that time will drag on and they can make major gains in the midterm elections. If that fails and bills like these pass, their only fallback plan is to continue the judicial coup, this time against both the executive and legislative branches of government. And that really might be a bridge too far.
No Sanctuary for Criminals is particularly explosive because it threatens to sanction sanctuary cities. The judicial coup currently stands in the way. But a judicial coup against Congress and the White House would be unsustainable.