DNS and pointing and SEO on temp site

Ok,
I have a website which is a wordpress site and is in a folder of my main server, with the domain pointing to it from a different domain registrar
So the set up is
http://www.server.org/websitename which is hosted on siteground
with http://www.websitename.com pointing to the above folder from a 123-reg account
it is featuring in top 3 on google with http://www.server.org/websitename showing, as we have done lots of SEO
the pointed websitename is ideally what we want to feature in google.
So, how do we change the DNS so that the domain websitename shows up?
I can buy an addon slot with siteground that then lets the websitename point to the folder I think
But will the strongly SEO’d site still feature with its actual path? And will google penalise for duplicate content
Or, do I properly install the websitename on a different server, and duplicate the site? With the proper domain name pointing to it, and hope that the SEO will work, although much offsite SEO points to the original server route.
Any advice welcome