Your question: "How blatant the circumvention of the Constitution has to be for SCOTUS to act?" indicates some confusion about the big picture of how contesting the constitutionality of a law works.
SCOTUS doesn't proactively do anything.
The Supreme Court cannot simply review a law that has been enacted and say it is unconstitutional of its own accord, or at the request of someone involved in the political process (some countries allow this, the U.S. does not).
The U.S. Supreme Court is not equivalent to the institution of a "Constitutional Court" found in many countries. It is just the last court of appeal for all U.S. Courts. It often ends up resolving constitutional questions, but only after other courts have already done so in cases where there are real tangible immediate consequences to the decision.
A lawsuit must be brought by someone who is actually injured for the courts to act
In your example, nothing would happen unless a home owner could show that soldiers had actually commandeered his home without consent or compensation, or places him in imminent fear of having this done.
If someone can't show that, then no lawsuit to determine the constitutionality of the law is allowed even if it is blatantly unconstitutional and the question of the law's validity will remain unresolved by the courts.
This limitation is called "standing" and requires that there be an actual case and controversy with a suit brought by someone who has suffered a legal injury before anyone can bring any lawsuit.
In point of fact, there are all sorts of laws in the United States that are clearly unconstitutional, but which are never brought before the courts to declare unconstitutional, because the government agrees that those laws are unconstitutional and makes a point of not enforcing those laws.
All cases (with exceptions not applicable here) start in trial courts
Suppose soldiers do commandeer Bob's house at the express direction of the President without Bob's consent or following any procedure that amounts to due process. What does Bob do?
Bob brings a lawsuit against the soldiers and their commanders up through the President and the United States in the U.S. District Court for the state where the house is located or where the defendants live.
Suits against the U.S. and its employees must be brought in federal courts rather than state courts.
SCOTUS can hear cases as a trial court, but only in cases involving a state or foreign country or a diplomat as a party (and in practice, even those cases are referred to a temporary judge called a special master for evaluation and SCOTUS only considers the case after receiving a recommendation from the special master). None of those circumstances apply in this case.
A federal trial judge hears the case and decides if the law is constitutional or not, and if it is held to be unconstitutional may decide that Bob is entitled to a remedy.
There will also be other separate issues to decide in the case. For example:
Was the lawsuit brought within the statute of limitations?
Are the soldiers immune to suit for damages against them personally, which depends upon how clear it was to the soldier that he was acting unconstitutionally?
Were the soldiers violating orders or following orders?
Did Bob meet other procedural requirements during the course of a lawsuit (like making the proper disclosures of information and showing up to hearing he is required to attend, and presenting evidence in accordance with the rules of evidence)?
If the trial judge finds that the law is unconstitutional, the trial judge can issue an order saying so and that is the law of land that binds the parties (including the U.S. in any other case presenting the same issue under a principle called collateral estoppel) unless someone appeals the case.
Every state and federal judge in the United States from small claims court judge to a U.S. Supreme Court justice has the power to declare laws unconstitutional if it comes up in a case properly heard in that judge's court, not just SCOTUS.
SCOTUS (with exceptions that don't apply) doesn't hear direct appeals
A handful of cases are directly appealed from a trial court to SCOTUS (mostly election law cases). But the vast majority of cases, including this one, would go to an intermediate court of appeals first.
If someone does appeal the case, it goes to the U.S. Court of Appeals for whatever circuit the state of the District Court is located in. It reviews the judge's ruling in light of the evidence presented and can either reverse the trial judge's decision or affirm it.
Only after the U.S. Court of Appeals has ruled (sometimes with one more layer of decision making within the U.S. Court of Appeals), any party can appeal the case by a writ of certiorari to the U.S. Supreme Court.
SCOTUS often declines to reconsider Court of Appeals Rulings
The U.S. Supreme Court doesn't have to take the case and 98% of the time that cases are appealed to it, it doesn't take the case. If it doesn't take the case, then the U.S. Court of Appeals ruling is the law and that ruling is binding on any other federal court in its jurisdiction in future case.
The U.S. Supreme Court will usually only take the case if it feels the decision was wrong, or there are conflicting precedents that have to be resolved from different courts.
Whichever judge decides constitutionality (a power not reserved to SCOTUS) that judge will try to follow the law to make the right decision whether the violation of the constitution is blatant or subtle.
If the U.S. Supreme Court does decide to take the case, it can affirm that U.S. Court of Appeals ruling (which is then binding on all U.S. Courts as precedent), or it can reverse the U.S. Court of Appeals.
In each case, at the trial court level, at the U.S. Court of Appeals level, and at the U.S. Supreme Court level, the only question is whether the law conflicts with the constitution as interpreted by the case law already decided over time.
Only a handful of cases in the history of the United States have ever squarely addressed whether a law violates the 3rd Amendment so there isn't a lot of directly applicable precedent, but the judges would also consider how similar provisions of the constitution, like the 5th Amendment, have been treated and would consider law review articles and historical records about the intent of the Third Amendment as well.
Judges have quite a bit of freedom in interpreting the law, but will try to rule in the way that most fairly represents what the total body of the law and interpretative information about the law says in the context of the facts before it.
In this case the government would probably lose but you can never be sure
In a case as clear as your example, the Government would very likely lose although no case is entirely certain, because it allows action at any time even though it is not a time of war, does not consider the home owner's consent, and does not create any meaningful procedure for exercising the right.
But, it really doesn't matter if the violation is blatant or if it is subtle. The court even routinely rules that laws are unconstitutional not because they actually violate a provision of the constitution directly, but because they merely "burden" the exercise of a constitutional right. A law that effectively nullifies a constitutional provision would usually be invalidated. Sometimes lawyers informally and in private call an argument that is technically valid (for example, by creating a procedure albeit a meaningless one) "too cute." Arguments like that usually lose.
The U.S. Supreme Court routinely invalidates laws that violate the constitution only in very subtle ways (e.g., requirements that have been interpreted to pose minor barriers to voting could be held unconstitutional), and the U.S. Supreme Court now and then refuses to invalidate laws that seemingly blatantly violate the constitution (e.g., "In God We Trust" on U.S. coins).
Often a non-constitutional or settled constitutional law question is resolved purposely in a way that avoids the need to rule on an unresolved constitutional question
Often, constitutional cases are resolved on the question of standing, or whether the right person has been sued, or by interpreting a law in a manner that is unnatural, in order to avoid having to address the question of the constitutionality of the law itself.
For example, in your case, a judge might say that "at any time" in the statute, really means "at any time during a war", and that "regardless of the objections of the owner" really means "over the unreasonable objections of the owner", and that there is a duty to pay fair market value for the use of the house under the statute because the law is silent on that point, even if none of those things, in a cold plain reading of the statute would seem to be reasonable interpretations of its plain language. And then the judge might say that interpreted in this way, the law is constitutional, but the government violated the law and the court might then award a remedy to Bob, because the government violated the law so interpreted rather than because the government enforced an unconstitutional law.
But, if it decides to take up a constitutional question because it isn't satisfied with how the U.S. Court of Appeals resolved the issue, it won't hesitate to do so.
