On the 1099-DIV you should enter the Payer's name. This will match the payer TIN.
On the other hand, you should record the Broker's name for the 1099-B. This will identify who issued the 1099-B.
The IRS will not be concerned about the broker's full name ... until they have a question about it. It takes just a moment. I'd suggest you enter the name as it appears on the 1099-B.
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