Is it ethical to share a controversial reason for declining to review with other editors/journals?
I think that the journal needs a written policy on what they retain and it should be visible and obvious to all, especially all reviewers. A request to review can/should point to the policy. 

Then there should be no issue at all here. If a reviewer wants to make a statement about why they refuse they know beforehand whether it will be public or not. 

If an editor gets a reason that they think should be handled differently (outside the policy), then it can be negotiated with the reviewer. In an extreme case, the review, itself, could be rejected. But a notice of "outside the policy handling" is part of the stated policy as well. 

If you make it clear and flexible there won't be an ethical issue.