I had the same problem. When I purchased my spinner from Tru-Turn they sent me the same prop nut/adapter that you have. This combo requires you to ream the prop out to almost 1/2". At the time I was using an APC 15x6 prop and they worked fine with this nut and adapter. When I switched planes and wanted to try the APC 16x4W I could not use this nut/adapter combo. The hub on the 16x4W is much thinner.
I had to switch to the nut/adapter that you have linked to on Horizon's website. This combo does not require the prop to be reamed out like the one you now have. Going to this combo solved my problem.
Another option is to use a safety spinner and light lock.
http://www.horizonhobby.com/Shop/ByC...odID=HIGSPN008
http://www2.towerhobbies.com/cgi-bin...?&I=LXL742&P=0
Eric