Congratulations on getting a very nice truck camper!
As others have noted, it probably can be done. We've all seen lots of rigs similar to yours.
Is your truck short wheel base? If so, your COG is also probably off since that camper wasn't designed to be on a short wheelbase. Otherwise, I think you can likely get a short extension. You want it as short as possible both to decrease the leverage on the hitch and also to make it less likely to drag. The side benefit is that it will be out of your way by only pulling the ball mount if it doesn't extend past the camper bumper.
You really need at minimum higher capacity tires to give you any margin for tire safety. You don't say what your current tires are. E could be 3300 or 3760. That would make a huge difference. Similarly those tires at 80 lbs have a lot more capacity than 65 lbs. Weighing is the only way to know how much is on the rear tires.
You also have a bit of a warning if your wife wanted you to slow down. That should tell you that you are on the edge. I can tell you my wife likes our F450 much more and volunteers to drive it. I can't remember her ever wanting to drive the SRW. Take that for what it's worth. In my case, the brake improvement and turning radius improvement is dramatic.
By the time you buy 19.5 wheels and tires, you probably could trade for a DRW. If you can't get by with a DRW, get at least the 19.5 wheels and tires. I drove my same camper with a 2006 Ford SRW, Extended Cab 4x4 PSD with 19.5s for over 8 years. Why so long? It was my daily driver and a DRW would simply not fit where I needed to park every day. However, you are way over your GVWR which is even less than mine was. My GVWR was 11400 and they improved the brakes and suspension in '05. Reasonable braking definitely will be an issue.
I believe the general rule of thumb for boat trailer tongue weight is 10-15%. You are at 5% if you think it is 250 lbs. If you want your eyes more wide open, go to a scale and weigh your rig with the boat attached.