not sure what version you're using, but in mine the 1099 only asks for the payer's name, not the business name.
The 1099 is then linked to Schedule C. On Schedule C is where you'll find the place to enter the business name but note no business name is required if there is no separate business name (that seems to be your situation).
As long as it's the same business activity the 5-year period for the hobby loss rule is not reset.
