Unless they changed the abilities of the doppleganger, it doesn't matter what weapon you use, it still can use your weapon skills.
I killed it in ??? seconds (never kept track - but longer than 5 seconds, thats for sure). I had a hammer, pure hammer skills (I knew he had a bow and an axe, so I thought, no chance for it). Sadly, he pelted me with hammer skills using an axe

. Yes, he attacks faster (since its an axe) and was able to use all hammer skills.
So unless they did change it, sword skills used by an axe are still viable for the doppleganger (but not for you).