Hi There,
I currently manage all of the Analytics and Tag Manger at on of Australia's largest Magento Partner Agencies, so getting this set up for you isn't a problem at all.
One thing that I do need to point out through is that you will never be able to get perfect sync between the two platforms - there will always be a slight discrepancy (Usually under 1%) simply due to people who have Java switched off.
To get this under way all I need is access to your Magento, Adwords, Analytics & Tag Manger account.
One question however - Have you tried tracking through Analytics instead of Adwords conversion tracking? The benefit of using Analytics based tracking is that you get accurate data in your Attribution reports which are fantsatic at giving you truely detailed data for your reporting!