Help UK Startup here. Client from U.S. won't pay

Hello, I run a publishing startup, and we are owed over $10,000 by a major american brand. They give us really good ad rates and I don't want to burn a bridge but they've only paid us once since Summer. If we get shirty with a lawyers letter they could probably say "fine, sue us" and we wouldn't be able to do anything as we are in the UK and they are in the states, we are a tiny startup with very limited funds-we have tried numerous phone calls and emails every single week since Christmas. Any help gladfully received! (Note: I can't reveal their or my startups name for obvious reasons, but am a 1st-time founder who really needs advice)