Carb ice

" ...........when an intercooler is mounted, because airbox temperature will be close to the outside........"

I very much doubt that the intercooler will be so efficient it will bring the inlet air down close to ambient.

Speculation; the chances of carburettor icing, in a turbo charged system, must be close to 0. 

"Sorry, but the temperature under the cowling is really important."

Of course its important but in a properly designed cowling, with adequate air flow (in flight), it is relatively unimportant and may, in a naturally aspirated engine, help to reduce the chance of carb ice.

You have taken logical/reasonable steps to lower the air temperature entering your turbo charger. The result has been reduced airbox (carb air) temperature which should see benefits in engine power & fuel consumption.

Sean, you can expect an intercooler to half down the temperature rise at maximum power. At idle, flow throught the intercooler is much less so the temperature decrease is better as at max power.

The rise by pressure at idle is zero because there is no pressure increase, good air in to the charger is at outside termperature.

Thereby the temperature rise at idle is only done by the warm charger itself, which is not a lot at idle.
I see that after engine start, where sometimes I do have carb icing during warm up.

At all this in summary, there will be the airbox temperature at idle without an intercooler about 4 to 5°C. With intercooler you will have 1-2°C which is not sufficient the preven carb icing. Due to this in the Rotax installation manual carb heating is recommended when using an intercooler.

With the statement temperature under the hood is important, I was speaking of inlet temperature for the turbo and in this case, I obviously hab about 40-50°C air under the hood at flight close to the inlet.

But result is always: without an intercooler you usually don't have to worry. With intercooler you have to think about it.