I can't say for sure but I believe that the max tow package also includes heavier rear springs as the GVWR move to around 7700 lbs depending on model. That impact would be obvious on the payload but no bearing on towing capacity which is more to do with engine power and trans/rear end gearing.
I THINK the off road package is the killer. I just tried to build my truck on the Ford website and it said the MAX trailer package is not available on the trucks with the off road package. Since I have the FX-4 package, it looks like that might be the difference.
I didn't do an extensive study, but it appears that the dry weight of the AK2612 looks to be 400 - 500 lbs heavier and therefore the CCC is less by the same amount compared to other TH's the same length.
I believe that the towing limit on your truck is listed at 9100 - 9400 lbs depending on configuration which is less than the GVWR of the camper. Between that and the high tongue weight, you'll be at/over the limits for the truck.