but with Germany's ridiculously overpowering air presence

Well there is problem number 1! They didn't, at any point in Summer and Fall Germany enjoy even air superiority let alone air supremacy over England. Limited action time, and losses that were irreplaceable in the long run contributed to mean they were never doing anything but trading 1 for 1 with the RAF and that was a losing proposition for the Luftwaffe. That means any attack on shipping would be opposed, and the best way to protect attacks on shipping is to keep your enemy from getting aircraft up in the first place, returning again to the issue of attacking Fighter Command's airfields.

But the greater issue is where the Royal Navy concentrated, that being the North of Britain and at the extreme limit or beyond the reach of most German aircraft. The great fleet base of Scapa Flow in the Orkney's had served as refuge for the Grand Fleet in WW1 and did so again for the Home Fleet in WW2, and the same with Rosyth and the Firth of Forth on the East coast of Scotland. The capital ships meant to oppose any German invasion, including the battleships Rodney, Nelson, and battlecruiser Hood were all based there at various points. While smaller squadrons under the Nore Command and Western Approaches were made up of destroyers and a handful of cruisers with other squadrons based on the Humber, Thames, and at Harwich.

And destroyers if handled well are notoriously difficult to hit by air attack with reliability unless you use overwhelming numbers. The Royal Navy lost about 50 destroyers to air attack during the war in European waters, that is a lot, especially as most occurred before 1942. But it also represents only about 1/5 of the destroyers on strength or building in 1939, and those losses are for the entire war remember. While most of those losses occurred while said ships were tied to a location where they were made more vulnerable, so Dunkirk, at Malta, or off Crete. If they can be in a position to get to sea in response to the sortie of a German invasion fleet, steam through the night to converge under cover of darkness it puts them in a much safer position.

Would also page /u/thefourthmaninaboat here and suggest looking at any of his awesome posts that popped up when the movie Dunkirk came out.

Firstly, the German air presence over the Channel was by no means overwhelming. Throughout the summer and autumn of 1940, the Royal Air Force (RAF) was capable of contesting the Luftwaffe's presence over the Channel and the South of England. During the early stages of the Battle of Britain, the Luftwaffe attempted to destroy British convoys running through the Channel. While RAF losses were heavier than they would have liked, the Luftwaffe's losses were the same. Over the course of July and August 1940, the RAF suffered 150 losses, almost all suffered protecting British shipping and ports. In the same period, the Luftwaffe suffered 286 casualties. The Luftwaffe's best weapon against shipping, the Ju-87 'Stuka', was also its most vulnerable. It was a slow aircraft that had to fly straight and level during its attack run. During this part of the battle, and during its later stages, losses were high amongst Ju-87 squadrons, especially when unescorted. While the British did, ultimately, reduce convoy transits through the Channel, this was to avoid adding an unnecessary risk to shipping.

More importantly, however, the Luftwaffe was fundamentally unprepared for an anti-shipping campaign. The majority of the Luftwaffe was poorly trained in anti-shipping operations. They had been trained to attack larger stationary targets - bridges and factories - or area targets like military units. Ships were a much harder prospect to attack. They could make radical manoeuvres to avoid a bombing run, and tended to have their own anti-aircraft armament that could dissuade an attack. The ships that the RN planned to use as its first line of defence against attack were destroyers. While most RN destroyers had a poor AA armament, they were highly manoeuvrable. Larger ships, while less able to take evasive action, had much heavier AA protection. We can look at the Luftwaffe's record against shipping at Dunkirk to give some suggestion as to how effective they would have been in a hypothetical Operation Sea Lion. During the course of Operation Dynamo, the RN deployed 40 destroyers. Of these four were sunk by air attack and a further six were heavily damaged by the same cause. In addition to this, the French Navy deployed nine destroyers, with one sunk by aircraft. While this represents a loss of a fifth of the force deployed, this loss ratio would still leave the RN greatly outnumbering the Kriegsmarine in the Channel. In the summer of 1940, the Kriegsmarine had just 10 destroyers to its name. The RN's Nore Command, the Command which was protecting the likely invasion areas, had 32. Subtracting a fifth of these leaves 27 destroyers available - still nearly three times as many ships as the Kriegsmarine had available. Edit: However, this estimate is skewed by the fact that Dunkirk gave the Luftwaffe the best possible conditions for attacking ships. Most targets were stationary, and in enclosed waters, preventing them taking evasive action. Of the four British destroyers sunk, all were stationary or moving slowly when attacked. The Luftwaffe didn't have the best record at identifying ships - during Operation Wikinger, a Kriegsmarine destroyer sortie in February 1940, the Kriegsmarine squadron was attacked by Luftwaffe aircraft, sinking the Leberecht Maass. In the ensuing confusion, another destroyer, the Max Schultz, struck a (possibly German) mine.

Finally, on a more technical level, the Germans lacked an armour-piercing bomb capable doing serious damage to the largest British ships. They would not introduce their most effective armour-piercing bombs until later in the war, and as such would be forced to attack the larger British ships with high explosive bombs or semi-armour-piercing bombs that could not effectively sink an armoured target.