It is impossible to know how likely this was, since it is very difficult to deal with historical counterfactuals, and the political situation in Italy at that point was sufficiently dicey that Clement VII had plenty of incentives not to accept a petition for annulment at that stage, as his relationship with Emperor Charles V, Catherine's nephew, was not great.

As the Roman Catholic Church took, and still takes, a rather dim view of divorce, attaining an annulment by appealing to Rome was still pretty much the only option for Henry if he wanted to remain a Roman Catholic in good standing. As you allude to in your question, Henry argued during the King's Great Matter that his original marriage to Catherine was invalid on Old Testament grounds; Catherine was originally married to Arthur Tudor, and it was unclear whether or not they had ever consummated the marriage. What made this even trickier is that Henry married Catherine under a papal dispensation in the first place. The status of Catherine's marriage to Arthur had a large enough asterisk next to it that the church had to OK it to make sure everything was canonically acceptable. These questions came up again when Catherine had trouble producing healthy heirs, and some interpretations of these Old Testament passages in the Pentateuch had been interpreted to mean that not having sons would be a penalty for incest. Henry VIII's moral scruples have not enjoyed much positive press, and with good reason, but he genuinely does seem to have come to believe that he was being punished for transgressing God's law, and that the original papal dispensation itself had been granted improvidently. It's hard to escape the logic that Henry's reasoning was largely self-interested, but however he reasoned it, he does seem to have come to have believed this at some point. Rome's position in all of this, of course, was that as a matter of canon law, Popes should not overturn dispensations issued under prior Popes.

Royal annulments, while not necessarily common, were also not unheard of. Henry's own sister, Margaret Tudor, had obtained an annulment from her second husband, the Earl of Angus, a move which Henry himself, ironically seems to have opposed at the time. There is also the rather strange affair of Anne of Brittany, who was married by proxy to the Holy Roman Emperor Maximilian I, then married King Charles VIII of France in 1491, who was already technically engaged to Margaret of Austria. The marriage was officiated quietly, and then approved later by Pope Innocent VIII, who annulled the marriage between Anne and Maximilian after the fact. Papal approval was needed anyway, as Charles and Anne were marrying within ordinarily prohibited degrees of consanguinity. As Kathleen Wellman puts it:

"Confidence that popes could be relied upon to grant royal annulments was warranted. Despite the sacramental, permanent character of marriage according to Church teachings, a royal marriage could be set aside by the pope in the interest of any number of secular concerns: Popes were interested in making this process available to foster their own political agendas; European monarchs wanted a means to set aside infertile wives or, on occasion, wives whose political advantage had diminished. So, while an individual king might object to another king’s annulment, as Maximilian did, or a pope might resist a particular royal annulment, all had a vested interest in the practice." (Wellman, Queens and Mistresses of Renaissance France, p. 72)

While it's impossible to know whether Henry's request could have been considered reasonable under different circumstances, there was precedent for royal annulments, and they were not always carried out under the most auspicious circumstances, as in the case of Anne and Charles, in which one party was already legally married, and the second marriage was conducted illegally prior to any kind of dispensation. None of this, of course, means that Henry VIII was a swell guy, but he wasn't just dreaming up fantastical nonsense when he tried to annul his marriage to Catherine

Readings

Christopher Haigh, English Reformations: Religion, Politics, and Society under the Tudors

Kathleen Wellman, Queens and Mistresses of Renaissance France

Jane E. A. Dawson, Scotland Re-formed, 1488 - 1587