It sounds like the primary purpose here is residency definitions, which could mean that the US would be obligated or entitled to consider the Bermuda company american for tax purposes.
IRL, that Bermuda company is likely to be a shell company, receiving payment from a single client (EG msft) so that the parent company can book it as an expense.
That said, it's pretty much impossible to say anything at all before details go live, and accountants have had time to chew on it. Tax rules are all detail. Headlines really do mean nothing. "Where they do business" could mean a lot of things. Sales. employees, manufacturing, financing/listing.
Corporate income tax is applied to the net of revenues and expenses.
Some of the language in other statements seems to suggest there will be some sort of a quota system. MSFT will be defined as x% american, y% Bermudan, etc. This is pure speculation though.
"Also, Bermuda is unlikely to agree to this."
If you are a simple, single entity, your income tax is calculated there. It's hard to say what if anything will actually come of this. Tax legislation has a tendency to hide the main point. But currently, small businesses disadvantaged by the things this legislation ostensibly wants to curb. Multinationals have the resources and complexity to avoid tax entirely. It's usually just sole traders that pay full income tax.