Religious groups, not just JWs, but Catholics, Orthodox Jews, Mormons, and various Protestant ones, would soon challenge such a new law, if the bill passes, on the basis of the long-established clergy-penitent privilege. This is what happened in Montana and why the WTS won. State legislatures and other law-making bodies can pass all the laws they want, but once they're challenged in court, it's up to the courts to decide how the legal conflict between two opposing principles would be handled.
The only way all this can be ultimately settled in the USA is for SCOTUS (Supreme Court of the United States) to take up the matter. Given the current make up of SCOTUS, full of conservatives as it is, such a change from the status quo does not look promising. From all indications they would simply declare the Washington law invalid upon the basis of the clergy-penitent privilege enshrined in federal jurisprudence.
I've made this point before in similar threads. You can get all excited about these gestures politicians make to please their constituents, but it's the courts that matter. There is a much better chance of change on this issue in Westernized countries outside of the USA.