Getting the Med strategy to work is difficult because you need an early Italian victory to get Malta, but you also need an early Italian defeat to persuade Italy to ask for German help! I'm not sure how you can get both.
It is a tricky POD to write. You need a series of POD or a POD with a series of butterflies to do a Med Strategy. It is easy to get a POD where the Axis take Malta at some point, or at least attempt to. And any taking of Malta without very heavy losses gives small plus to the Axis. And this will add several months delay to the Allies clearing North Africa and may well extend the war by weeks. Not really a big deal, but that is what we get when we do a secondary theater and a secondary battle POD.
And then there is a related topic of a Med Strategy, which people normally mean "a strategy to knock out or cripple the UK in WW2". This is an entirely different beast, and the King points are largely correct. And I think it takes an early POD than he suggests in many ways. Lets go back to WW1, where the Germans/Austrians had a 2 front or 4 front war depending on how you look at things. They could have easily beat either Russia or France. And Falkenhayn and to a lesser extent Hindenburg focused on knocking out small powers (Serbia, Romania) and then focusing on one power at a time. In the WW1 environment, Russia out of the war and only fighting France and the UK at the beginning translates to a curb stomping of the Entente. So in many ways, Hitler achieve with a stroke of a pen what the Kaiser/Falkenhayn/Moltke could only dream of. And it WW1, with Russia out of the war and France conquered, the war would be basically over, even if the USA decided to join at some point. This attitude was taken into WW2, and there was little serious, high-level thought given to how we finish off the UK after France. A lot of the the issues with not taking Malta, or Wiking's various ideas for an airwar with the UK, or OTL North Africa reflect this attitude carried over from WW1. IMO, it is almost mandatory that you have a simple POD that is horribly complex to write due to butterflies. You need some competent naval officers focused on the UK to join the Nazi party and have a stature of the same ballpark as Goering. It will not get you a larger budget, but it will get you a focused plan, and more importantly, it gets you to start the ground work. IOTL, by the time the Germans had overcome the shock of "my god, France fell that fast" and started to come up with a strategy, the UK had recovered from "OMG, France fell that fast" and they too had a plan. Luck is where opportunity met preparation. The fall of France provided the golden opportunity of the UK focusing on saving England with lightly defended Malta and other areas. The Germans were not prepared, at least organizationally. If if quite crude and basic, the Germans having a War Plan Take Out UK drafted by the navy and approved by Hitler by 1/1/1939 goes a long way towards fixing the issues.
You also have the issues of Italy. I think it is also solvable from the perspective of writing a TL. And I see the POD being more in Germany than Italy. Germany had a lot of contempt for Italy, and IMO treated Italy badly in the treaty where France surrendered. Once you have the POD where German takes the Med seriously or defeating the UK before France is invaded, you have several routes to accomplish the desired effect.
1) Closer co-operation between Germans and Italians Navy after Munich. You can lay the seeds of Italy/German having a plan to handle the situation in the Med here. Now Hitler likely needs a personality transplant, but once that happens it is doable.
2) After the Hitler is surprised to see France and the UK enter the war, he can realize that help is very useful. Again, without a POD going way back, it requires a personality transplant for Hitler, but to look at the effects. With say October 1939 to April 1940 to get a plan together, you could have a competent Italian entry to the war.
3) Now to more what the thread is asking for, you have a situation where France falls and then we have the "OMG, France fell so fast", but we have a competent, rapid response. If you are serious on taking the UK out, you have Italy at the table and you get the demands on France focused more on how to take the UK out. You get less reparations, better treatment of POW and you get concession that help the naval war. You get serious about trying to figure out how to get the French Fleet in Axis hands without the UK being able to sink it. (Yes, this is very hard since UK gives max effort). You concentrate on basing rights in Africa. You concentrate on France not just going neutral but switching sides until the UK leaves. We don't really think about it much, but you could have French forces actively help fortify West Africa or western North Africa. We call it a Med strategy, but a Med strategy does not take the UK out of the war. You need a "Shut the Suez, shut the South Atlantic" strategy to win unless you want to take 4-5 years to do it. And with Italy at the table, you can give Italy more, and get cooperation. The details can vary, but think Tunisia, Corsica, and Chad for Italy in exchange for being OK with some German help. And Italy gets to have Suez after it is capture.
4) Now to what the thread really ask for. We get to July 1, and we try to figure out what to do. It takes months for a multi-million man organization to switch focus. Overall, the Germans adapted pretty well to the changes until they went into Russia. And like I started with, sure Malta falls. Maybe you can bribe/threaten Franco into letting you attack Gibraltar. You can probably find a way to use the Eastern Libyan ports not Western with lower losses than OTL. Maybe you can be strong enough to get the Allies to cancel torch, and if so, Italy is a huge winner, but the war does not really last that much longer. Probably 2-4 months. If not, you will have the USA pouring low quality but well supplied divisions into western North Africa and the Monty commands grows in power each month. The end is inevitable, even though you may well delay the process long enough to have the USA not invade Sicily.