They are not in the same Jeep, so I can't make a direct comparison on how well they cool head to head.
The BC1193 looks to be better made than the Cold Case. The workmanship and quality is much better on the Champion. The fitment was relatively the same. Not that the Cold case is a bad radiator, I just like the quality and performance of the Champion better.
Here's what I can tell you about the performance. On my wife's 97 trail rig, I originally ran an American Eagle 2 row radiator. AE is Champions sister company. That radiator was great, never ever got over 210, no matter the outside temp, A/C on, trails, or on the street. Unfortunately, that radiator sprung a leak, and AE no longer makes that one. So I opted to replace it with a Cold Case. It's just okay, it creeps up just over 210 fairly easily. When it's warm outside, my wife says she has to toggle between running the A/C and using the fan override switch. It's manageable, but it's not as good as the AE.
On my 99 trail rig, I tried the Mopar stock one. It worked okay for a time, but after a while, the temps started to creep up. I couldn't really tow my teardrop trailer in hot weather without over heating issues. I think the issues may have been exacerbated by a faulty heater core letting gunk into the system. So I bypassed the heater core, cleaned everything out, and put in the BC1193. So far it's worked really well. Still have a little bit of temp creep when towing, but nothing like the stock Mopar. The BC1193 has worked the best for me.
I've tried other radiators like the CSF 2 row. That one was junk for me. On my 95 trail rig, I had a custom radiator built for it. I had the biggest tanks I could fit made with a 2 row core. I even moved the whole assembly closer to the front of the grill to accommodate not only the bigger core, but for the ZJ fan clutch. Even that one just works okay.
All in all, on all of my trail rigs, the fan override switch helps the most to mitigate the temps. If you shut off the A/C and run the e-fan, all of them will cool way down. Fortunately, I don't need to run the A/C all the time, and most of the time on the road, I can run the A/C no problem. It's towing or on the trail that can give me issues.