Hillbilly Hauler

If the tongue weight is ok, which it probably is because of the counter weight of the bikes on the back of the trailer, I think you will be fine. Maybe not enough tongue weight?

Its easier on gas with the weight on your back vs pulling the extra weight. jmo

As someone already mentioned get or have made a receiver extension. Get the camper up into the bed as far as possible. not so much as whats behind as getting some weight back on the front axle. Easy enough to move the bikes fore or aft for tongue weight.

Putting the camper in the bed of the truck will not work with this trailer. I think the rails would hit if I went up a hill while turning.
If I take the tailgate off there wouldn't be any support under the camper. As mentioned brakes would be good but no time to do it so I'm only taking one bike.
The receiver extension would worry me more than towing without brakes. I've got time tomorrow to get the tongue weight figured out.

So what will you be doing with the camper when you get there? Will it stay on the trailer?

The camper is staying on the trailer.

OK.

I think as long as you tie everything down good you'll be good.

If the camper wasn't staying on the trailer, I'd likely have put it in the truck, and got another smaller trailer for the bike that would work with your hitch & tailgate situation (bit longer tow pole?).

Another small trailer with a longer tongue would be great but this is a short notice kind of trip and I'm making due with what I have.

I had a friend do almost the same thing, but he slid the camper to the back over the axles and loaded the bikes on the front, under the overhang and forward of it. Worked real good for him for many years.

Craig

Do not attempt without trailer brakes. It looks to me like you have way too much weight in back. If you have lights, brakes, load secure and good hitch you should be ok. Check that tongue weight. I've done a bit of towing and understand what is needed for a safe tow. What you show does not look safe to me.
Many states have brake requirements for any tow over 1000 lbs.