trying to remove the rear bumper w/ the brackets still installed is difficult to do with the bed still in place. the brackets are angled & hit the bed when trying to remove to bumper. you said that the replacement bumper didn't have any brackets on it, this is the reason. just remove the plastic cap to gain access to the top bolts. soak bolts w/ some PB. replace if they are junk. undo plate light harness, unbolt bumper & remove. @ this time, if you need to do so, swap the harness or plate lights. clean/paint if desired. install new bumper & bolts, align before fully tightening. install cap/s, enjoy the improved looks.
I live in Mass, lots of rust & rot. I can remove these bumpers w/ a good 6pt socket on a 1/2" breaker bar w/ a good 6pt wrench on the back side to break it free, then swap to long handle 1/2" ratchet to finish it up.
some people don't have the luxury of a compressor & a gun in j/y. besides doing it by hand is your exercise for the day. You can honestly tell your spouse that you worked out. seriously though, its not difficult