Yes the hyd oil cooler and the radiator are two seperate coolers but the are one unit, so you have heat transfer.
I think your missing the bigger problem and if you can isolate which system is the main culperate then I think you can start to fix the problem.
First, on your hyd temp. It needs to be below 180 degrees. If your cooling system can not maintain 180 or below it creates a condition I call "run away". If the temp goes above 180 you can not control it, it will just keep climbing till an alarm goes off or causes damage.
Second the engine cooling system. I though they should run below 200. Not sure and I know each engine is different. Double check what your engine coolant target temp should be.
Now as you might have seen the cooling "problem" is already brewing. If your hyd side is trying to keep the temp below 180, the engine side is running at 200, the head transfer will heat up the hyd side till it reachs 200. As the hyd side temp raises from the heat created by the engine side, the engine side raises. and back and forth. I bet the engine temp should be @180. This would make them work together.
So, now you know why the dedicated machine's run seperate, indipendant systems. Now your job is to try and figure out which one is getting hotter first. You could take a cheep meter and put a probe on each outlet and monitor temp. I suspect your "radiator" is clogged with dried up dirt in the corner's. I suggest removing the cooler and washing with a high psi pressure washer. I like the thermostat change as well.
I bet with this info and if you perform some of these test you'll find where your problem is coming from. These little machine's are a pain!!
I was in Pensacola Wed. Small world.