In a word, "No."

In several more words, here's an extended extract from a letter that Lawrence wrote to Sykes in September 1917 from Lawrence's authorized biography by Jeremy Wilson. My notes, addenda, and clarifications are in brackets [] below, all other parentheticals () are Lawrence's. I've also separated out some of the longer paragraphs for readability:

You say they [the Arabs] will need French help afterwards [after the war] in the development of Syria— but do you really imagine anyone in Syria (bar Christians) wants to develop Syria? [NB, "Syria" here means at the very least the area encompassing both modern Syria and Lebanon, it's not clear to me whether or not he intends to include Palestine and Transjordan but in context in this instance I suspect not, though this entire area is in theory part of "Greater Syria"] Why this craze for change? A slow progress, utilising only the surplus resources of Syria itself, seems to me more desirable than foreign borrowing, and a forcing-bed of public enterprises; and I think this point of view will be uppermost in the minds of the Damascus Government.

If the French put a tangible price on their help, the Sherif might pay for it. He is not going to sell "spheres of influence" etc. for gold or mountain guns [which the French were now offering, in theory without a price, but implying an obligation]. If the French want to annex a province, in return for "X" materials, let them say so. You can't buy gratitude by a secretly conditional gift.

I see you [Sykes] saying "pre-Fashoda" again! But please remember that Fashoda occurred before my time, and so the gibe is a generation late. [In his previous letter, Sykes described the maximalist positions that would set Britain and France at odds as "Fashoda-ism" in reference to the pre-war Fashoda crisis over competing French/British claims that intersected around South Sudan.]

The Sykes-Picot Agreement was made before the Sherif revolted. It stipulates for the erection of a sort of Arab-Albania [I have no idea what he means here]; and resigns to France a great deal of what British people in the Near East regarded as, at least, mutual area. The agreement was based, apparently, on the idea that the Allies were going to destroy Turkey, and strip her of her foreign possessions in their favour. We thought you had been generous to the Arabs, and were told unofficially that the need of bolstering up French courage and determination in the war made it necessary to surrender to her part of our own birthright.

This latter statement, if true, justifies our suspicion of French action in the Near East. They are to get something of ours out here in return for their help in France. If we show an anti-French bias, it is your fault, for not educating us in Egypt officially to the knowledge of how much the French do require to stiffen them to last out the war in the west.

Then the Sherif's revolt took place, and what had been a generous attempt to revive an Arab state in Syria under the aegis of France and England, became the sphere which the Sherif might obtain if he succeeded. You observe that we gave him no reward for his efforts on our side. He might take— what we had already given. [NB, as I'll explain below, I don't think this is quite true.]

Therefore I don't think the Sykes-Picot [Agreement] can stand as things are. The Sherif will succeed, given time and a continuance of our help: he will take by his own efforts (don't assume virtue for the mules and cartridges we supply him: the hands and heads are his) the sphere we allotted to our foreign-advised "independent Syria", and will expect to keep it without imposed foreign advisors.

As he [the Sherif and by extension the Arabs] takes this sphere of his, he will also take parts of the other spheres not properly allotted to an Arab state [...] and what are the two Powers [England and France] going to do about it? It seems to me that England and France can either take their areas first, or turn the Arabs out by force, or leave the Arabs there, or leave the Turks there (by ceasing to pay the Sherif's subsidy. It will be quite impossible for whoever is the military adviser of the Sherif to advise him to leave a B area [of French influence, I think he's actually mis-written this here and means the A area marked on the map] or a blue area [of French rule] alone, if the Turkish army is using it as a military area against him: and military occupations have a queer habit of developing into political ones.

[...] You know i am strongly pro-British, and also pro-Arab. France takes third place with me: but I quite recognise that we may have to sell our small friends in pay for our big friends, or sell our future security in the Near East to pay for our present victory in Flanders. If you will tell me once more what we have to give the Jews [i.e. what would become in a couple months the Balfour declaration in negotiations that were already known to be ongoing], and what we have to give the French, I'll do everything I can to make it easy for us.

Feisal is as reasonable as a soon-to-be succesful man can be, and now is the time to mould him to our wishes. The future seems to me all over thorns, since military action by the Arabs, independently, was not in our minds when the Sykes-Picot [Agreement] was made [again, I don't think that's really true as I'll try to explain], and if it is to be a Mede and Persian decree [i.e. inalterable], we are in rather a hole: please tell me what, in your opinion, are the actual measures by which we will find a way out.

Phew, so to summarize, in Lawrence's view, Sykes-Picot had been negotiated in a wartime context that had not considered the success of the Arab revolt. He was already at that time in a position to see that the Arab revolt under Faisal would have taken direct possession of the territories allotted to the French under Sykes-Picot. Critically, and I think this is something misunderstood about British imperialism generally, he raises no real objections to those territories allotted equally to the British in part because, as he says earlier in the letter from a portion I didn't quote that "The Sherifs accept British help openly, since they believe it disinterested (gratis, if you like.)

Gilbert Clayton, who was to deliver the letter to Sykes and was Lawerence's superior in Cairo, read the letter and decided not to deliver it to him. His reply to Lawrence is just as interesting for how I imagine Lawrence and others involved in the Arab revolt and Palestine Campaign must have regarded the situation at the time of writing:

From all I can hear the Sykes-Picot agreement is in considerable disfavour in most quarters viz: [Lord] Curzon [who served on the War Policy Committee and was made Foreign Secretary after Balfour in 1919], [Charles] Hardinge [Permanent Undersecretary at the Foreign Office], D.M.I. [the Director of Military Intelligence], D.I.D. [the Director of the Intelligence Division] etc.... I am inclined...to think that it is moribund.

[....] The Sykes-Picot agreement was made nearly two years ago. The world has moved at so vastly increased a pace since then that it is now as old and out of date as the Battle of Waterloo or the death of Queen Anne. It is in fact dead and, if we wait quietly, this fact will soon be realized: it was never a very workable instrument and it is now almost a lifeless monument.

To explain the two references I made above about not considering military action at the time, Lawrence seems to be taking Sykes-Picot as prior the Arab revolt, which is true, but is ignoring that it was after, and indeed largely the product of, the Hussein-McMahon correspondences. The entire point of the Sykes-Picot negotiations was to understand what the extent was of French reservations that the British mention in the Hussein-McMahon letters, whose entire aim was the emergence of a state of war, in one form or another, between the Arabs under the leadership of Sharif Hussein against the Turks. That the Arabs would secure what territories they did was actually a conditional provision of the Sykes-Picot agreement.

What is certainly true is that at the time, Lawrence and others and Cairo could not have imagined the way the Arab revolt would play out. They were generally in favor of massive landings at Alexandretta to block a critical supply gap in the Turkish rail lines, and the operations at Gallipoli were still under way (though Alexandretta was not to be and Gallipoli was failing.)f The other explanation, one that seems, in one degree or another, quite likely is that in 1917 Lawrence didn't quite understand what Sykes had done or what Sykes knew at the time. That was one of the problems from the outset, by the time Sykes concluded the negotiations with Picot, he, and he alone had been fully briefed on the contents of the Hussein-McMahon letters and the negotiations with the French and the wishes of the British cabinet and the India office. It would be a difficult task for any man, and Sykes was by all accounts a pretty mediocre diplomat. Lawrence, while despite being a junior officer, was apparently consulted on the Hussein-McMahon correspondence, but neither he nor his superiors in Cairo were consulted prior to the finalization of Sykes-Picot, which is actually pretty remarkable given that by all rights McMahon ought to have had a major say.

I might try to add more to this, as there's more to say on the question and I only received this book a few days ago and it's excellent, but sourcewise this is from the Authorised Biography of T.E. Lawrence, which seems to cover this and a lot of similar questions I've had myself for some time in very precise detail.