I'm not an attorney but I don't think Congress can just demand that laws be ignored; they have to change those laws. So, if the Antiquities Act isn't amended then there'd be a legal conflict. I think it would end up in the courts right away as the Department of The Interior would not be able to carry out such a law if it would then also be violating another.
We'll see I suppose but I really don't think this is going to go anywhere; just a symbolic gesture.
It's not a case of other laws being ignored, it's more a case of which law has the greater power (i.e. takes precedence over another law).
I'm fairly sure that Federal laws take precedence over State laws. [link]
Also, a newer Federal law (especially one where "Notwithstanding any other provision of law or policy to the contrary ..." is specified within it) will take precedence over an older one. The old law will then only apply to the point where it does not conflict with the newer law. The only thing that would 'outrank' the newer Federal law would be the Constitution, if the new law was seen (believed) to be unconstitutional then it would go to the Court to be resolved.
Regrettably, not a symbolic gesture. If the HR7608 bill becomes law [and it's already passed the House] it will be very real. Any inaction or failure by the National Park Service to abide by the terms of Section 442 will leave them liable to prosecution.