The only "peace" Hitler the corporal would accept is unconditional surrender (like every other country Germany overrun). Until then, the rape and pillaging would continue
Except that's blatantly false... France was given terms, even if not very good one. And with the British in a much less assailable position than the frogs... Well, with invasion almost impossible, fairly lenient peace terms are the only way to get the poms out of the war.
The RAF would be removed. As we know from history the RAF received a break. Germany stopped going after the RAF, and started with city bombing. As we know you must air superiority, so keep after the RAF. Once you get rid of the fighters, now you can carpet bomb the cities. Even Churchill wanted to kiss the Germans for bombing his cities, and giving the RAF a break. Plus they stopped bombing the radar, another corporal move
A pop-history myth... Even before the shift to attacking London British fighter production and pilot training was more than keeping up with their losses, while the Germans were struggling to make up their losses. And that's with only 11 Group Fighter Command heavily engaged...
Once Sea Lion starts Germany can move far more supplies into the UK, than a few ships coming in from overseas for the UK. So just one or two "lucky U-Boats is all that's needed to slow down the UK supplies. And Germany has most all of Europe to draw supplies from
Except the British didn't just have 'a few' ships coming in... they had tens of convoys a month each of dozens of multi-thousand-ton-dispalcment ships arriving and departing each month. Against that a 'lucky' U-Boat or two is nothing...
Meanwhile, the Germans' need to rely on two thousand river barges with very limited cargo lift, landing cargo over beaches and through badly damaged small ports in the face of surface attack by the Royal Navy.
Which one do you think can actually shift more supplies?
Now as for the over stretched navy ..... many things can be done. Much of the UK navy was not in home waters. Bombers are an option. Mining both ends of the channel is another option. Shore artillery and naval ships are other options.
Per British forces in home waters... Multiple capital ships, more than a dozen cruisers and in excess of a hundred destroyers plus numerous lighter vessels. Opposing them? No operational modern capital ships, one clapped out pre-dreadnought, one operational panzerschiff, all of three cruisers and less than a dozen destroyers...
IOTL the Luftwaffe had a couple of dozen
float plane torpedo bombers equipped with rather dodgy torps as their only dedicated anti-shipping force. Anything more requires diverting the 300-odd Stukas which are also suppose to be doing a million other tasks at once...
Mining the Channel is possible but doing it on the scale to 'seal' the Channel requires a large number of mining operations by warships. Which is liable to tip the poms off to the plan and is also liable to loose you more than a few minelayers.
Most coastal guns don't have the range to engage ships the other side of the Channel, and OTL use of coastal guns against slow British merchant convoys in the Channel proved rather ineffective. Against a large force of fast warship? Not gonna help much at all.
German naval ships? Please see comment on numbers above.
Edit to add:
You make very good points. As for the navy, no question German is very out numbered. Germany capital ships however are newer, well built, and can out gun (range) most UK ships. That clearly makes a real cat and mouse game.
S&G were out of action after taking torpedo and Destroyer hits off Norway. Bis commissioned in September 1940 and was still working up and undergoing trials for months thereafter. Tripitz didn't commission until the end of Feb 1941.
So, no German capital ships available...
Per gun range... theoretically, yes (). In practice, however, without either extensive aerial spotting or radar spotting (not really functional until 1942-43...) hits rates beyond about 25,000 yards were so abysmal that this nominal advantage can be effectively ignored.