Ok the question revealed the answer (clarifying that NAME is not dimensional). The solution that seems most clear is something like the following. Note I'm also joining another table D that only joins on A.ID to demonstrate it must come after the joins on B,C.

Please scrutinize.

with NAME as (
  select distinct A_ID, NAME from B
  union
  select distinct A_ID, NAME from A
)
select distinct a.ID as 'A_ID', b.NAME as 'B_NAME', c.NAME as 'C_NAME', B.etc., C.etc., D.etc.
from A a
  inner join NAME n on n.A_ID = a.ID
  full join B on a.ID = b.A_ID and n.NAME = b.NAME
  full join C on c.ID = c.A_ID and n.NAME = c.NAME and (b.NAME = c.NAME or b.NAME is null)
where (b.NAME is not null or c.NAME is not null)