Orsdxy

Negotiations over the current US government shutdown have, as far as I've heard, mainly been between the President and Congressional Democrats. So far, Republicans in Congress seem to be siding with the President in refusing any agreement not including a border wall, but there seem to be some cracks in this.

Suppose that a sufficient number of Congressional Republicans decided to break with the President and reached an agreement with Democrats, but which the President still found unacceptable. As I understand it, both houses of Congress could pass a spending bill along those lines. The President might then veto it (he could stall for up to 10 days first). Suppose, however, that Congress had the votes to override the veto (2/3 of each house).

If they were to override his veto, would this end the shutdown, or would the President somehow be able to continue it anyway?

I wonder if there is any argument that, even if Congress allocates money for the Government, it is up to the President to decide whether to actually spend it.

If Congress has the 2/3 votes to override a Presidential veto, they can pass any budget they want with zero consideration for what the President thinks. Ever since the Impoundment Control Act of 1974, the President no longer has the authority to refuse spending Congressionally allocated funds.

Therefore Republicans are free to end the shutdown by agreeing not to allocate funds for the Mexico Wall and obtaining the required number of votes from the Democrats. Likewise the Democrats could agree to fund the wall and obtain the necessary votes from the Republicans. Which side to blame for the shutdown is up to you.

If they were to override his veto, would this end the shutdown, or would the President somehow be able to continue it anyway?

This would end the shutdown.

I wonder if there is any argument that, even if Congress allocates money for the Government, it is up to the President to decide whether to actually spend it.

Congress appropriates money for specific purposes. Unless Congress specifically delegates authority to someone else in the Executive branch, every dollar is supposed to be spent as stipulated in appropriations legislation. The President choosing to do something else with the money (including not spend it) would be illegal and likely be subject to a court challenge, and has been in the past.

It is also worth noting that as a purely tactical matter, there would be no value in trying to continue the shutdown in some sneaky way in the event of a veto override, because that much popular support in ending a shutdown in that fashion would mean that the President would never get what he wants from even his own party in Congress. The only reason the shutdown is continuing now is under the theory that some number of Democrats would agree to give him something he wants (e.g. if not the stated wish for $5 billion dollars for "The Wall" then some less significant but still desirable thing).