Is it permissible for me to alternate between praying the Sunnah (supererogatory prayer) of Asr and sometimes omitting it, based on the premise that it is not a confirmed Sunnah and on the action of the Prophet, peace and blessings be upon him, knowing that I have vowed to consistently pray all the Sunnahs for every obligatory prayer?

The regular Sunnah prayers are divided into two categories: emphasized (which the Prophet, peace and blessings be upon him, consistently performed) and unemphasized (which he sometimes performed and sometimes omitted). The Sunnah prayer before Asr (afternoon) belongs to the second category. If you vowed to consistently perform the emphasized Sunnah prayers, then the Asr Sunnah prayer is not included in your vow. However, if you vowed to consistently perform every regular Sunnah prayer established from the Prophet, peace and blessings be upon him, then the Asr Sunnah prayer is included in your vow, and you are obligated to consistently perform it. If you are unable to fulfill the vow due to illness or similar reasons, then you are required to offer an expiation for a broken oath.