I use callipers for inside measurements BUT they probably aren't accurate enough in this case to directly compare the bore of the adaptor to the dia of the spindle.
Chucking the silver steel in the three jaw and grabbing the free end with the collet chuck should show errors in alignment of the mount and the collet taper. To improve the test chuck a larger piece and turn the free end down to fit one of the collets. The turned end should show zero runout, when you mount the collet chuck to this runout at the end has to be error in either the collet or chuck. If the bore runs true the problem has to be with the fit of the mount. Turn the free end down to fit a smaller collet, if the error changes the collets may be at fault, if the error stays the same the chuck is at fault.
Chucking the silver steel in the three jaw and grabbing the free end with the collet chuck should show errors in alignment of the mount and the collet taper. To improve the test chuck a larger piece and turn the free end down to fit one of the collets. The turned end should show zero runout, when you mount the collet chuck to this runout at the end has to be error in either the collet or chuck. If the bore runs true the problem has to be with the fit of the mount. Turn the free end down to fit a smaller collet, if the error changes the collets may be at fault, if the error stays the same the chuck is at fault.
Free advice is worth exactly what you payed for it.
Greg
Greg