The Bank has their Closing Costs, and the Buyers have theirs. I doubt that the Bank is asking you to pay for theirs.
If you have an Approval/Acceptance, you are way ahead of the game.
Follow your Realtor's advice and get your house.
Your Escrow will probably take 45-60 days, during which time you will get your financing in order and do your Inspections. Everything will be fine unless you find something really big with the Inspections. If it's something small, accept it and keep going. You're almost HOME.
Good luck and may God bless