France wouldn’t allow it and the British themselves wouldn’t want it
Well, that actually depends. Britain would want it, it was just that getting it would require more than they were willing to pay. There are actually a number of places that you could change this.
Sykes-Picot discussions began because the British were beginning to consider crossing the Egyptian border into Palestine. This was an area of interest for both nations so it was thought that an agreement was necessary to avoid problems. However, the French, AIUI, did not have a definite set of war aims for the Ottomans Empire. Vaguely, if the Empire was to be carved up (which wasn't guaranteed in British or French thinking at this stage) then France would want Cilicia, Lebanon, and probably the coastline in between. It was only when the British came to them that they realized they had the leverage to ask for more land. In some ways Sykes-Picot was a bit of a steal for France. They were able to be considered an equal partner in an area that had been won mostly by British troops and local allies the British had done the negotiating with. If the British go into the negotiations determined to get Syria from the start (or just forget to consult the French as I understand it was a bit of a last minute scramble by diplomats as they realized the implications of the military's plans) then at this stage there is a good chance that they can get France to agree to Syria being in the British sphere in exchange for some concessions on French business interests and support on other French gains.
France did have significant investment in the area that had only started to be challenged by British and German investment in the run up to the war. So they will still likely want it. But if the British come in with bigger demands, the French can, I think, be bargained down to allowing Syria in the British sphere. It would help if this was all decided before 1916 when it becomes obvious that the OE is starting to break down and France is starting to give more importance to its colonial possessions as their economic salvation post-war.
If Britain does get control of the area then Faisal likely remains king of Syria and I think it is very possible that the British later unload Jordan to him (though probably not for free). Lebanon would be difficult due to French interest in the area, and Britain taking control of it might be a step too far. Palestine is also likely a sticky issue. With the Balfour declaration in place I am not sure if any agreement including Palestine in an Arab kingdom would be good enough. Without it, there is still an interest in international control of the area. It is possible international control could eventually break down and the area be transferred to Hashemites though?
Alternatively, there is the McMahon-Hussien correspondence which was going on at the same time as Sykes-Picot. somewhere along the way , it was realized that the demands of the Arabs and French were in conflict. There was enough grey area in both agreements that the British thought they were technically not in conflict but they more or less knew that one or the other would call foul if both were adhered to. It was considered that all three parties representatives should get together to work out the points of contention, but it was believed there was not time (remember this was supposed to be preliminary to an attack into Palestine). If the groups had all been brought together it is possible (though not guaranteed) that a solution working for everyone
might have been achieved. Again, doing this earlier reduces French demands.
In any case, I think the best option for creating an Arab kingdom as defined in the OP is to do it at the negotiating table with Britain and France involved. If you have to weaken these powers to the extent that they cannnot intervene you have probably changed things to the extent that you have to worry about others doing the same thing.
