bike rack

That reduces a 2.5 hitch to accommodate a 2 inch draw bar or other accessory.
Clicky

Look at the wall thickness of your bumper. Welding onto the thin material could cause some issues.
Your spare hangs pretty much over the bumper. A four bike rack will hand in back of the bumper which will exert a lot of twist on the bumper.

http://mobile.walmart.com/#ip/REESE-Towpower-Hitch-Box-Reducer/31954785?type=search tv

This is what I have. I was planning on reinforcing it somehow. Mind u 2 bikes r small kids. Was also taking spare off. What's the difference in this and the receiver clamp things they make. This way seems stronger to me but I do understand u still have the bounce factor

Many of the 4 inch square tube bumpers are not strong enough for two let alone four bikes.
You can get a generic hitch to attach to the frame that utilizes a 2 inch receiver style rack. Something like this one listed for MH but may fit trailer.
Multi-fit hitch

It all depends upon how sturdy the bumper is welded and/or reinforced. Under most situations, the majority will tell you "No! Don't do it!" Their reason? Even if the bumper is strong enough to hold the weight, the excessive G-force on the hitch when the camper is bouncing down the road causing the bicycle rack to bounce with even more force could cause the bumper to come apart.

Then there are those that will tell you if you can reinforce the frame and weld a hitch to the frame and bumper, that is the best way to do it.

Then there are those that will tell you if you do the welding, you run the risk of invalidating any warranty you may have on your camper frame. (which is true).

Then there are some that say, "No problem! I've never had a problem with bicycles on my back bumper!" I am one of those!

All three of my Travel Trailers had the spare tire mounted to the rear bumper with no extra support I had to fabricate. On my first 2 campers I attached a bumper mount and attached my bicycle carrier to it. Never had a problem (Maybe I'm the exception?). My third and current camper also had the tire mounted on the bumper, but the bumper is factory reinforced with a smaller hitch. I am not carrying the bicycle any more, but do carry a basket - cradle now and removed the spare tire completely. I don't think the weight is as heavy as the bicycle and carrier now.

I think, unless you reinforce the frame, 4 bicycles is a bit much with the added weight of the carrier itself. That's a lot bouncing on the back end. One bicycle you may get away with, but I don't think the bumper will survive 4 of them. I REALLY do not think any of my bumpers would survive 4 bicycles. But when I do carry the bicycle, I have only the one since our kids are all grown and gone now.

First Travel Trailer:



Second Travel Trailer:





Third Travel Trailer:



I suggest you get with a good welder who knows his business and have him inspect the integrity of your camper frame and determine the best way to weld reinforcement and the receiver.