I don't think it would be wise to add this to your existing setup.
I would not add to the rear bumper, not designed for the type of stress that 4 bikes will place on it and could twist the bumper right off. Also a lot of weight so far behind the trailer axle could upset the balance.
You could look at the A-frame mount from Arvika (LCI has a cheap knock off). They make bike specific ones, but mine is the simple 2" carrier so I can use our Thule rack on the trailer or the truck.
As shown in the picture, it just cleared the back of our SUV on turns with 2 bikes, and now with a pickup I have a couple of divots in the tailgate from too tight turns. With 4 bikes though it would probably interfere with the vehicle. You could also easily mount it rear facing with a pup so the bikes extend over the top, but I would worry (a lot) about whiplash and the bikes colliding with the top of the pup.
The cleanest way to 4 bikes on a pup would be cross bars. Very stable, not very expensive.
Doesn't help that you already have the rack, but this worked great for us for the 6 years we were in a pup.