You have the rating on your JAR.. As long as you fly within JAR world, you can use your privileges for flying the N thing.. Check your FAR 61 for detailed infos. However I strongly recommende to advise your insurance...
Am not totally sure CL300 , you can normally fly a "N" reg A/C on behalf of your JAA Licence, as long as you stay within the national airspace of the licence issuing country... If you have a JAA -UK delivered- CPL/IR , then you can apply full privilege in UK Airspace, not outside . I may be wrong .