Specifically, it's that due the structure of the US government, it's nigh-impossible for the US to make binding agreements at all, especially in one-off cases like this. Our diplomatic representatives are under the Executive branch, which is subject to change every 4 years, and successors have no obligation to uphold agreements that their predecessors made. The power to actually make law that binds American citizens falls to Congress under the Legislative branch, which has to pass law making any agreement the diplomats make legally binding internally, which they may or may not do, and the diplomats negotiating such terms have no power to sway that one way or another. Then the actual execution of the agreement would fall to the courts, which could potentially find a deal that managed to be negotiated by the diplomats and legislated by congress contradictory to the Constitution, and nullify it.
Add on top of this the difficulty the US has passing any legislation whatsoever, and you've got a whole mess on your hands.
And I almost forgot that the accused would likely be facing charges in an individual state, which adds another layer to the mess, wherein the state likely has the sovereignty to execute one of its residents regardless of any international agreements that the Executive branch of the federal government makes.
Add on top of this the difficulty the US has passing any legislation whatsoever, and you've got a whole mess on your hands.
And I almost forgot that the accused would likely be facing charges in an individual state, which adds another layer to the mess, wherein the state likely has the sovereignty to execute one of its residents regardless of any international agreements that the Executive branch of the federal government makes.