Obviously, a treaty cannot circumscribe the constitution. No treaty can change the president's term from 4 years to 3.
The constitution defines the president as commander in chief. It also defines what powers of the military congress can regulate. I think we can agree that a treaty cannot regulate the president on war more than passed legislation can.
As to UN treaties and NATO treaties on who we can attack, I think it is clear that those provisions are void IF we recognize that the CinC power includes the ability to launch attacks in the name of national emergency. I would say that power to launch attacks, while I don't agree with the common interpretation, is recognized as legitimate. If that is the power of the president, and congress cannot regulate (as every president has rejected the war powers resolution) then a treaty cannot.
Nuclear weapons are funded, so it isn't a "power of the purse" issue.
That leaves the geneva conventions, similar agreements, and the ability of congress to regulate tactical and strategic decisions.
I have seen the full range of views on this from
-congress has the power, if it chooses, to direct the most tactical level decision (this law directs platoon A to take hill Wolf 359).
-Congress has no power to even direct strategic decision (what funded weapons to use when).
When we think of if laws can restrict the president as CinC, my bet is the court would say these laws are nonjusticiable and leave it as an impeachment issue.
Well, the president can be impeached for anything. However much risk he wants to tolerate, Bombs Away.
But then we are back to the Joint Chiefs, and they would be confined by those agreements, if we recognize a power to regulate strategic decisions is held by congress.