You're reading way too much into this "pursuance" language. Its only purpose was to make clear that ordinary statutes passed under the old Confederation do not have to be considered the supreme law of the land. It's a given that laws and treaties must not conflict with the Constitution, because it's the Constitution that creates these powers in the first place. You're trying to make a hyperlegalistic argument in order to deny the obvious.
And I found it rather interesting that earlier in the thread, you were demanding that I find a SCOTUS opinion that supports my position, and then when one was presented to you, you blithely claimed that you know better than them anyway. Just shows how determined you are to close your eyes and ears to reality.
Southack confuses what the word "laws" means. It doesn't simply mean "statutes", it encompasses all acts passed by Congress, which includes the treaties. That is why Article VI refers to the Constitution, the laws and the treaties, collectively as "the supreme Law of the Land". The term "law" has a far broader meaning that simply "statute".
Nevertheless, I believe that the President's conduct in regards to the illegal combatants is both prudent as well as substantially constitutional.
--Boot Hill