The agent should pay the transaction coordinator, or do the paperwork him/herself. You should ask the agent to pay the $350 for you. That’s how it usually works – several in my family are agents.
Or better yet, walk on the deal and get an agent that won’t nickel and dime you.
Nancy_s is right – $350 is .1% of your purchase price. But consumers get screwed in real estate by lots of similar fees. Don’t give in! It may be a small percentage of your price, but wouldn’t you rather have a couple dinners out for that $350? Or make your car payment? Or get a hotel room in Santa Barbara?