Generally speaking, shortening the prayer while traveling is a legal concession that portrays Islam’s tolerance and simplicity in matters of worship. The Prophet used to shorten his prayer whenever he was on a journey. He (peace and blessings be upon him) is reported to have said: “Allah likes His servants to undertake the legal concessions given to them in the same way as He likes them to observe their obligations.”
While performing Hajj, you are a traveler who is allowed to shorten his Prayers. You are also allowed to combine Zuhr with `Asr and Maghrib with `Ishaa’.
As for performing sunnah Prayer while traveling for Hajj, we’d like to cite for you what Sheikh Sayyed Sabiq, states in his well-known book, Fiqh As-Sunnah, in this regard:
“The majority of the scholars are of the opinion that it is not disliked to perform nawafil (sunnah prayer) during the state in which one is shortening his Prayer. On this point, there is no difference between regular sunnah prayers and other nawafil. Al-Bukhari and Muslim record that the Prophet (peace and blessings be upon him) made the ghusl in the house of Umm Hani on the day of the conquest of Makkah and then he prayed eight rak’at.
Ibn ‘Umar reports that the Prophet prayed while riding in whatever direction he was facing and nodding his head [i.e., for the movements of the salah (Prayer)]. Al-Hassan relates: “The companions of the Prophet while on a journey performed supererogatory prayers before and after the fard salah.”
Ibn ‘Umar and others are of the opinion that there are no nawafil, before or after the fard salah, except for during the middle of the night. He saw some people praying after the salah and said: “If I were to pray, I would have performed the whole salah [as obviously that would have taken preference]. O nephew, I accompanied the Messenger of Allah [on journeys] and he never prayed more than two rak’at until Allah took his soul. And I accompanied Abu Bakr and he did not pray more than two rak’at.” He also mentioned the name of ‘Umar and ‘Uthman, then he recited the verse: “Ye have indeed in the messenger of Allah a beautiful pattern (of conduct).” This is related by al-Bukhari.
Ibn Qudamah combines what al-Hassan and what Ibn ‘Umar say by concluding that al-Hassan’s report points to the fact that there is no harm in praying nawafil while traveling, whereas Ibn ‘Umar’s report points to the fact that there is no harm in not praying such nawafil.”
So, according to the majority of Muslim scholars, shortening Prayer because of traveling for Hajj does not prevent performing sunnah Prayers.
Allah Almighty knows best.