You can join the three tables together and also do a text search on the name.
SELECT business.biz_name
FROM business,business_tags,tags
WHERE business.biz_id = business_tags.biz_id
AND business_tags.tag_id = tags.tag_id
AND (business.biz_name LIKE '%bank%' OR tags.tag_name = 'bank')
Will that do what you want?