The Original poster is completely correct. The BrE pronunciation audio-example does use an /e/. I am pretty certain that this is an error caused by the word being said in isolation and not actually being used. Strange things happen to our pronunciation when we read or when we say words in isolation. That pronunciation is definitely not one I've heard before. It is also not what is written in their transcription. The usual standard British pronunciation is definitely /ˈɜːɡətɪv/.
Another very interesting fact about that word is that the -ative ending in English is often pronounced with a KIT vowel, /ɪ/. So many speakers would say /'ɜː(r)gɪtɪv/ in both British and American English. If you listen carefully to the American example, the woman clearly says /'ɜ:rgɪtɪv/ not /'ɜ:rgətɪv/. Fun, eh?