First:Jesus drank nothing. The gall mingled with wine : they forced the stuff upon his mouth and he had to taste--- but it says, "just the taste and it was undrinkable. So yes, Jesus might have <~spit~> the TASTE from his mouth or rather off his mouth.
The second was a pure act of hateful contempt to humiliate Christ. The instrument used, was used by soldiers for much baser ~healing and purifying~. Christ could not be debased further through human means, but He "TOOK" it like only God could in his deepest descent into the hell of his Suffering for sin.
I would not suggest that the suffering of Christ was in any manner a part of the reconciliation God would look upon for sin. The BLOOD is what was to be shed, not the suffering. The suffering was the means that God chose for the blood to be shed, but EACH point of suffering there was blood letting. The bruising, the wounding, the scourging, all that pertains to the suffering of the cross was a manner of blood letting. It is the Blood that is to be the focus, for without the blood there is no redemption.
Second:
Though it may seem to some a small matter, Mark's account has the same person that got the sponge was the same that said the words of mockery.
Both Matthew and Mark state that the Lord Jesus drank. So there is no "rejection" or force the second time.
The first time there was an intoxicant added to the mixture. It was (according to some historical readings I did long ago) not unusual to prolong the torture and the crowd attention. But, Christ would never take an intoxicant into His being, then or ever.
The second time, was not an intoxicant, but vinegar. Vinegar was a common solution carried and used by the Roman soldiers to clean wounds, and sometimes used as a rub, or antiseptic.
There is no reading that Jesus "spit" this out, but that He drank.
Here is Mark's account:
When some of the bystanders heard it, they began saying, “Behold, He is calling for Elijah.” Someone ran and filled a sponge with sour wine, put it on a reed, and gave Him a drink, saying, “Let us see whether Elijah will come to take Him down.” And Jesus uttered a loud cry, and breathed His last.
Matthew's account only varies in that the mocking was done by others. Perhaps it is not disagreement between the accounts rather is showing how the mockery spread as ridicule usually well breads from one to another.
Here is Matthew's account:
Immediately one of them ran, and taking a sponge, he filled it with sour wine and put it on a reed, and gave Him a drink. But the rest of them said, “Let us see whether Elijah will come to save Him.”
Medically speaking, the second time, minutes before expiration, the typical sludge of the dying would have been rinsed so the final proclamation(s) of Christ could be both heard and understandable.
But folks don't have to take my word for it, they can well read the accounts and draw their own conclusions.