Unlikely to have legal teeth. For one, it's actually facially unconstitutional. First, the vesting clause of Article III states that the judicial power is vested in the Supreme Court. It also expresses that Congress may establish other courts. There's a statutory interpretation canon, expressio unius est exclusio alterius that essentially says that if you are speaking of a set of things ("courts") and say that something applies (congressional discretion) to a specific subset ("lower courts"), it implies that the remainder of that subset ("the supreme court") is exempt (cannot be regulated by congress).

Second, there's arguably already a constitutional process for regulating the ethics of SCOTUS ("good behavior"); however, the Constitution is silent on how it's enforced and in that vacuum SCOTUS's position is likely to be that they self-regulate.

I think, however, the public spectacle of it does have value. For one, SCOTUS (and the Republicans) are very concerned with the court losing legitimacy and a genuine consensus emerging that the court ought to be overhauled- whether that's Whitehouse's bill or a packing plan. It would be great propaganda for a future election for Democrats to say "hey, stop taking bribes" to justices currently taking bribes and for the court not to agree and say "you're right, no taking bribes" but instead to say "fuck you, you can't tell us what to do but also we're not taking bribes we promise".

Whitehouse's strategy here, especially since this would never pass the House, is to offer this as ammunition to Democrats running in 2024 showcasing the corruption of the Republican party.