I am WAY out of my depth on this one, but I seem to remember you first need to decide whether you want to take the amount as a lump sum or a periodic payment. I thought it made a difference. You do not report lump sum withdrawls on the UK return(?), but strict time periods between withdrawls are involved if I remember correctly.
If it's periodic, I would keep it simple. It is reported on the UK return (again if I remember correctly), and 90% is taxed by the UK. I'm sure there must be a much more clever way to do it, but I would pay the UK tax first and then claim FTC after declaring it on the US return, and wait for the refund. If you've claimed the entire $10,000 to the UK, then the refund may not have to be declared to the UK later (but I'm very unsure about this). While you're waiting, you can feel good about your benevolent tax free loan to help pay down the US deficit.
If you wish to go the first route, then the use of a pro tax advisor may be the best bet. Doing it the first way could become very complicated for future (or amended) returns. (And I do understand the difference in tax rates. Is it worth it for $10,000?)
Again, I'm really guessing on this one.
For the first route, if you declare the US witholding, I'm not sure you're only taxed on 90% of the amount by HMRC. The way the form is laid out, it may be 100% minus the FTC(?). Worth checking the instructions.