Just re-reading the original post and if I read this correctly you actually applying for a job with a UK company, as opposed to being sent there by an existing employer?
If you are applying for a job in the UK then you absolutely should apply as a British citizen because you may not get the job if you apply as a US citizen.
As a US citizen you will need a work permit which your employer will need to apply for, and they won't get it unless they can prove that they can't fill their vacancy with a citizen of an EU member state.