I encountered this problem a few years ago with the onset of presbyopia. Distant vision was and still is fine. I wanted to wear a pair of varifocal lenses with a near prescription at the bottom and clear glass above. Not allowed. I then asked about bifocal with near and clear. Not allowed. All that was permitted was a pair of look over (half moon) spectacles or a full lens near vision flip up/down type of spectacle. It was not allowed to have a pair of spectacles on a neck chain to put on as required (not enough time was the reason). This was for a Class 2 CAA PPL medical. The flip up/down was the best option for me. Bit of a buggeration all round.
Your AME does not seem to know the rules then, because the CAA clearly state that bifocals and progressive lenses are acceptable.
http://www.caa.co.uk/default.aspx?catid=49&pagetype=90&pageid=9245