No, it doesn't look like it.
From Pottermore, we see that the Sorting Hat is very stubborn about sticking to the decisions it arrives at. It seems that it's impossible to protest that the Sorting Hat got the decision wrong and to ask it to reconsider.
The Sorting Hat is notorious for refusing to admit it has made a mistake in its sorting of a student. On those occasions when Slytherins behave altruistically or selflessly, when Ravenclaws flunk all their exams, when Hufflepuffs prove lazy yet academically gifted and when Gryffindors exhibit cowardice, the Hat steadfastly backs its original decision. On balance, however, the Hat has made remarkably few errors of judgement over the many centuries it has been at work.
("The Sorting Hat", Pottermore).
Of course, the hat does sometimes take a very long time to arrive at a decision. As the question says, it frequently takes into consideration what the student themselves would like. Where the person and the hat remain in conflict, it seems likely that the hat makes the final decision, as was the case with Neville.
Of Harry Potter’s contemporaries, Hermione Granger and Neville Longbottom came closest to being Hatstalls. The Sorting Hat spent nearly four minutes trying to decide whether it should place Hermione in Ravenclaw or Gryffindor. In Neville’s case, the Hat was determined to place him in Gryffindor: Neville, intimidated by that house’s reputation for bravery, requested a placing in Hufflepuff. Their silent wrangling resulted in triumph for the Hat.
Whether the hat effectively chose on Neville's behalf or whether, after the hat dug its heels in, Neville finally relented and gave his consent to be put in Gryffindor is unclear. At that stage, Neville isn't exactly a strong-willed person who'd dig in his heels in a confrontation (when he gets older it'd maybe be a different story) so it may be that we don't have a clear example of someone who's truly prepared to stand up to the hat. Nevertheless, once the hat has shouted out the name of the house there doesn't appear to be anything the student can do about it. You have to go to your allocated house and make your peace with where you've been put. There isn't really much point in having a magical hat to decide who goes in which house if the decision is open to negotiation.
Even when it's a very tight toss-up between two different houses, the Sorting Hat sticks to its guns. It was still insisting that Pettigrew belonged in Gryffindor years after its original decision.
The only true Hatstalls known personally to Harry Potter were Minerva McGonagall and Peter Pettigrew. The former caused the hat to agonise for five and a half minutes as to whether Minerva ought to go to Ravenclaw or Gryffindor; the latter was placed in Gryffindor after a long deliberation between that house and Slytherin. The Sorting Hat, which is infamously stubborn, still refuses to accept that its decision in the case of the latter may have been erroneous, citing the manner in which Pettigrew died as (dubious) evidence.
So the hat's decision is final and there doesn't appear to be any way to challenge it.