If the black plastic part is what I am thinking it is, then I have been there and done that, when I was installing my tire carrier on a non-step stock bumper.
There are 2 bolts that are connected together with a piece of black plastic, this keeps them spaced appropriately to go into the frame horns so that a nut can be placed on them. It's hard to describe but the black plastic is inside the bumper part, but can't be removed since the bumper is not deep enough. My nuts were a bit rusty(
) and because the bolt head is encased in the plastic there is nothing to grip onto on that side, no amount of PB Blaster was going to help. Ultimately I gripped the nut with an adjustable wrench and twisted until the plastic broke apart, then I was able to get a pair of Vise Grips onto the broken plastic side and unscrew everything.
Once it was all apart I just went to the hardware store and bought the highest grade hardware I could and used an actual nut, bolt, and lock washer. It has held tight for a year now.
Here is a pic of the stock bolt and nut. As you can see the top of the bolt doesn't even have a normal hex head on it, since this part was inside the black plastic.
They other weird thing is that even if you get the nuts off, on a non step bumper you can't remove the two bolts and black plastic holding them together because there isn't enough clearance. So if you want to replace with an actual nut and bolt you have to break everything apart no matter what.