You would indeed need a different set of Axis priorities, but that's probably a good thing for them. But sending a mere 4 mobile divisions to Africa (the maximum which could be maintained) in September 1940 couldn't affect Barbarossa any worse then sending 2 Pz divs in Feb 1941 (OTL), indeed in those intervening months the Suez Canal could well be captured. Nor would sending forces to Africa in September damage the Blitz and Uboat happy time which limits Britain's ability to do anything about Rommel.
"Sending a mere four mobile divisions to Africa" acheives absolutely nothing unless those divisions can be supplied. These supply lines, assuming Gibraltar is not captured (which is borderline ASB anyway given the problems of supplying the paratroops needed to take it; there's no possibility whatsoever of an amphibious operation working against Gibraltar), must run through the Central Mediterranian, near Malta. Assuming the Italians even have enough ships at this point to supply these four divisions, which might or might not be the case, particularly after a couple of months, you still need to protect these convoys against British naval forces in theater and against British air assault from Malta.
From here it becomes a real puzzle. To supply a large army in North Africa you need to push the Royal Navy out of the Mediterranian. To do that you have to capture Malta, Gibraltar, and Suez (capturing Suez means capturing Alexandria as well, otherwise it would be included). But to get armies to Suez you have to supply them.
This puzzle can only be solved by ensuring a larger and fitter Italian navy, by the capture of the French fleet (and the training of reasonably proficient crews) upon the French surrender, and/or a German prewar fleet buildup and the placement of much of this fleet in Italian ports before the war. Now, it is very possible that the Axis-to-be navies can be increased in both material and effectiveness in the prewar period, but going simply off OTL naval force levels in theater during the period considered, a Mediterranian strategy for the Axis is simply not a war winner.
P.S. Are there any good "Fitter Axis Navies" TLs floating about?