Is there a tax excemption treaty that would apply to me as a UK resident who is doing a postdoc in the US on a J1 visa for two years? If so, how would I go about claiming that since currently tax is taken out of my paycheck (I am already in the US since 3 months)? Would it make any difference that I am a french national but lived in the UK for the past ten years? Thank you!

Sorry, but no. The 2001 tax treaty has NO SUCH EXEMPTION for teachers ot researchers.