Steady on posters....
Carb heat is both preventative, and curative, and should be used as required, by an informed pilot. The affect on engine power resulting from the use of carb heat, has the same variabilities as those induced by atmospheric change on the engine, because it is an atmospheric change, as the engine sees it!
If you're relying on the exact effect of the application of carb heat, both in temperature change in the carb, and affect on power produced, you may be disappointed. It's just not that precise. How much power does your engine make anyway? We really don't know... The design will have a rated power, but many factors, most certainly condition, will affect that. Even a newly overhauled engine is rarely dynomometer checked for actual power output.
Carb heat is always unfiltered, as it also serves as the "alternate air" which must not been drawn through the air filter. I have once used it, when I encountered freezing rain, the air filter instantly iced over, and the engine stopped. "alternate air" got it running again, for my retreat from those conditions. It had not been a carb ice event, but rather obstructed induction system.
I share the opinion that the pilot should manage power as required, and if this means moving carb heat from hot to cold for a go around, than that should be done. If the go around decision is so short that you just can't get to it, then fly the plane, and check configuration as soon as you can as you begin the climb away.