A sensible person would argue (correctly) both that order matters, so that the Constitution takes precedence over treaties in case of a conflict, and that treaties made outside the Authority of the United States are explicitly excluded by the qualifier "Treaties made, or which shall be made, under the Authority of the United States". The problem is that many who want power will pretend those issues do not matter, and we cannot trust even the Supreme Court to protect the Constitution.
And with the lyin’ kings take on this, all he needs is his signature to commence implementing, well, anything he wants with or without senate approval/RATification. It’s good to be the king. I guess.