It's hard to see how this would work if the system did not give somebody's judicial opinion the force of law. Courts wouldn't know what to do with it. But any unanimous Supreme Court decisions are not likely to be that controversial. In a controversial case, there's bound to be a dissent. So the system of overruling would just have to give that opinion the force of law.
So, I would suggest something like the following. Where the Supreme Court decision is 5-4, 2/3 votes of both Houses of Congress give the minority opinion the force of law. Where it's 6-3, you need 3/4. Where it's 7-2, you need 90%. Something like that.
If you're unhappy about giving Congress that much power, you can also require presidential assent.
In this way, you could overrule Supreme Court decisions without cluttering the Constitution with too many amendments.
Also the "rights" ideology which is the source of all this would be left in the Constitution. That document, as I said, should never have been converted into a work of sublime political or anthropological philosophy. It should have simply remained the rules for the functioning of the government. Until its highfalutin philosophical ramifications are removed and it is in fact turned back into a government rulebook we will never be safe from such social manipulation.
Constitutional amendments, from the very beginning, should have been limited to such things as adding a fourth branch of government, eliminating a branch, changing rules for qualification for any federal office, etc.