The hawk wasnt used until taka sasuke and the other taka members commented they had never seen it before the first time he used it. If he had it in the hebi days he would have used it in his fight with deidara since he was a sitting duck the whole fight. For long ranged attacks sasuke doesnt have much that wouldnt be negated by gaaras sand. Gaara literally specializes in long range jutsu i doubt sasuke would want to stay far away for long. As for kirin, it requires some prep. Imo kirin is sasukes best chance in the fight, but the likelyhood of him being able to use it is very low if they are fighting in a desert. Not only that but without a hawk to get around gaara has way better mobility with his sand.The fight will be a mid/high diff fight in Sasuke's favor. Genjutsu apart from Kirin does damage. The user casts Genjutsu on the Byagoku user and then he will put her to sleep. After that he will take the advantage. This is a regular genjutsu and the same he used against Sakura.
You must be registered for see images
Also Chidori Nagashi works if he gets within range.
The exact time reference to when he got the Hawk Summon isn't mentioned but I think it is only fair in ambiguous scenarios to give them their entire skill set according to that mode. While it's true that Sasuke doesn't have Deidara's Explosive KKG, he can simply invent it off his own. He can use Kirin as that explosive to damage Gaara's shield. Sasuke's Kirin does more damage than Deidara's C4 so if Gaara could barely stop his regular clay detonating device by making bigger clay birds what's to say that Sasuke can't do the same.
Gaara uses up a lot of sand from his gourd against explosives. That is a sign of his weakness. Since he is tiring himself out by blocking the bigger impact attacks, he is missing the smaller ones. Kirin can penetrate through that shield easily destroying it. Gaara is a long ranged user but so is Sasuke. He will at least want to keep a 5m distance to avoid being touched by his Chidori Stream. For long ranged users to battle against each other I don't see why the terrain causes a huge impact. I agree if you are a CQC fighter, you are at a disadvantage. Like for Deidara the landscape didn't bother him because he is a long ranged fighter. If Sasuke fights long ranged and then closes in the distance, Gaara's sand will not be able to keep up.
In my opinion gaara would win in the desert. Outside of the desert sasuke has a better chance. Deidara was barely able to beat gaara with a way to fly around, he lost an arm. Ive been arguing with the presumption sasuke doesnt have the hawk summon though. I figured that was the case since it was hebi sasuke, not taka sasuke. The fight would go differently if he could fly around. Being on the ground against gaara is very dangerous especially if youre in a desert.