A source contacts a journalist offering sensitive information with the expectation that their anonymity will be zealously protected. Then the journalist comes back to the source asking they risk compromising their anonymity further by talking to an entirely different news organization. This would qualify as 'dicking with sources' and no sane journalist would normally do this, for both professional ethical and business reasons.

What you're suggesting involves both violating the trust of the source and creating the impression the news organization needs its competitors to establish its credibility. I can't imagine this is anything short of a complete non-starter for Bloomberg, an actual serious news organization.