If they had gained air supremacy over the Channel, the German's first move probably would've been to start shelling the southern coast of Britain using a low-value naval asset. If the Brits send in a warship to stop the shelling, attack it with the Luftwaffe. If the Brits don't send a warship, keep on shelling with the goal of preparing a segment of the coast for the landing of German troops.

The Luftwaffe had about 2600 planes in 1940 and the ability to build as many airstrips as they pleased all along one border of the Channel. What do you think the chances were of a British ship or task force's surviving a single mission into the Channel? I agree with what you're saying that the Brits would've been willing to risk their entire Navy to stop an invasion of the homeland (which BTW was the original reason for the creation of their Navy centuries ago) but would that have been enough? (And if they lost most of their Navy, would they have been able to continue to feed their population? Remember that the Brits were dependent on food from overseas.)

If some power wants to attack a US aircraft carrier, they face the formidable task of finding it first. On the open ocean this is very hard: during the Cold War, the Soviets deployed an entire constellation of satellites for the sole purpose of locating US carriers in the North Atlantic (then transmitting the location to their submarines and long-range bombers). I mention this because the reputation for awesomeness earned by the British Navy before WW II was earned mainly in unconfined waters; against air power and subs, it is much less powerful in the confined waters of the Channel.

Also, look what happened to the British Pacific Fleet the next year when it dared to sail within range of Japanese aviation assets. And I think that happened in unconfined waters.