It's interesting you should post this as I'm in the middle of trying to fix the exact same problem.
I fitted one of the GTech fan shroud kits and wired it up as per the instructions so the fans came on automatically along with the stock IC fan.
I also wired in a manual control switch.
I then had the same problem you describe.
The fan warning light would come on after a few minutes of driving, seemingly only when the engine was hot, but the manual switch would turn on all 3 engine bay fans no problem.
After a bit of investigation I discovered a break in one of the wires going to the engine bay temperature sensor, right where the wire goes into the connector plug.
Once this was sorted the dash warning light stopped coming on and the fans would come on automatically as well as with the manual switch.
Now I've got what I think is a continuation of the first problem.
When I'm driving along the fans keep kicking in on their own for a few seconds every time my speed changes.
I can be sitting at a constant speed and it's fine but as soon as I touch the throttle the fans kick in for about 5 seconds and then go off.
This happens every time my speed increases from a constant but only for a speed increase, it doesn't happen when coming off throttle and I can't make it happen when sitting still and changing the revs.
I know it's not temperature related as I can be sitting at 70mph for miles and all is good, it's only when I touch the throttle do the fans kick in.
I can keep dabbing the throttle and make the fans come on at will.
This only seems to start happening when the engine is warm, it doesn't do it for the first 10 minutes or so of driving.
I know the fans are definitely kicking in because the idiot light on the manual switch lights up and at night my lights dim for a second due to the intial power draw of the fans.
It's particularly annoying at night as my lights are constantly dimming when these fans keep kicking in every time I change speed.
Any suggestions from anybody else about the above would be most welcome
Regarding your particular problem,
1) No, the front rad fans are separate from the engine bay fan.
2) The warning light on the dash comes on because the ECU is seeing a fault somewhere.
When the light comes on the engine bay fan/fans won't be working via the temperature sensor control.
3) Either a fault with the fan itself, a wiring problem or a faulty engine bay temperature sensor.
Regarding point 3, how have you modified your fans? I'm interested in hearing how you've modifed yours as we seem to be having similar problems.
You can try disconnecting the engine bay temperature sensor.
This should cause the engine bay fan to run constantly.
If the fan doesn't run right away with the ignition on and the warning light comes on right away it means you have a problem with the fan itself or the wiring.
If the fan does run with no warning light after unplugging the sensor then the problem might be with the temperature sensor itself.