Also, personnally I think giving away Middle East to Stalin is a good move.
Turkey will likely accept to join Barbarossa in a "War of Continuation" even if they resent Wagner for his earlier pact with Stalin. The Finns did exactly that IOTL.
More importantly, Stalin will have millions of soldiers either killed, or stationed in Persia, Afghanistan (a BIG quagmire), and possibly India.
He will also even less expect an attack from Germany.
Plus, of course, he will have to face the Germans ALONE. With full support from all continental Europe countries, and Japan hanging like an albatros near East Siberia (like IOTL, even if the IJA just sits there, it forces Stalin to maintain divisions east).
Britain and US may sell some weapons to SU to make profit but won't really support the Soviets, who will be much more feared and unpopular since they stabbed Britain during the last war. Plus British opinion will be weary of any entanglement in the war, and British elites hate communism.
More or less the same for the USA.
And of course, once Wagner starts liberating Ukraine, Bielorussia... and revealing Stalin's crimes, the world opinion (including British and US) will be tipped in Germany's favor. Ot at least, nobody will dare to suggest to support Stalin.
In a war with Germany, the Soviets are in a much better position with the British controlling the Middle East and providing resources through Persia road, than having to occupy everything themselves in Middle East.
So any advantage Stalin gains there would be useful in a peaceful world (access to Indian Ocean, no more British at his south, weakened Britain), but won't really help when Barbarossa comes.
Bases in Bulgaria won't really help Stalin that much either, since Bulgarian / German forces will corner them at the beginning of the war.
The only real advantage Stalin gets (in regards to Barbarossa) is full control of Finland (so no risk of Axis attack on Leningrad).
And even that is a double-edged sword, since Stalin still has to guard against possible German / Norwegian / Swedish / Danish attack in Finland. And to occupy whole Finland (quagmire).
Plus, those concessions will lull Stalin into false security even further. After all, if Wagner planned on betraying him, he would not give away whole Finland (road to Leningrad) nor bases in Bulgaria, right ?
The REALLY best scenario for Wagner would be achieving a total surprise against Soviet Union, by preparing massing his troops in Germany proper and moving them by train to Soviet border (with all their equipment). So Stalin would have days at most to suspect anything.
And then crossing the lightly guarded border (since the Soviets sent more troops, and the better ones, to Middle East) and blitzkrieging through Baltic countries, Ukraine and Bielorussia.
To keep the surprise, Wagner could choose to not tell any of his allies, and start with just German troops. And having Italians, French, Spaniards, Romanians... join only after the operation is started.
Maybe use the weakest forces (from the weakest allies) for logistical and garrison duties, while keeping the best forces (German or not) for frontlines, too.