Did any options exist for US draftees intensely opposed to the Vietnam War on political grounds?

No legal options existed for men who refused military service on purely political grounds during the Vietnam War. That said, by the late 1960s what was included as being "religious" conscientious objection to war could be very broad. Men who otherwise might have thought of themselves as being opposed to the conflict for principally political reasons sought, and were often granted, status as conscientious objectors.

The 1940 Selective Service Act allowed only those who were opposed "by religious training or belief" to "war in any form" to be exempt from combatant service. This was a change from World War I draft laws, which only allowed members of specific religious denominations (such as the Quakers, Mennonites, Amish, and Brethren) to be granted conscientious objector status. The 1940 Selective Service Act immediately created legal problems, because it was not clear what counted as "religious training and belief." In the 1948 Selective Service Act it was specified that "religious training and belief" had to be in "relation to a Supreme Being." This was an attempt to restrict conscientious objection to theists, particularly those who clearly belonged to historic peace churches.

Several groups, including the Central Committee on Conscientious Objection and the American Civil Liberties Union, lobbied to expand the definition of who could count as a religious conscientious objector. In the 1965 Supreme Court case United States v. Seeger the court found that an agnostic could qualify as a religious conscientious objector. Five years later, in the 1970 case Welsh v. United States the Supreme Court upheld the right of an atheist to be religious conscientious objector, even though he personally claimed he was not religious.

The expansion of the category of "religious training and belief" to include agnostics and atheists meant that almost any pacifist might qualify as a conscientious objector, as long as they objected to all wars. The Central Committee on Conscientious Objection put out pamphlets urging men to consider classifying their objections to war as "religious" when they filled out conscientious objection paperwork. Many men got conscientious objector status, even if they personally felt they were not religious believers. Often clergy were willing to help these men in their efforts to claim conscientious objector status from their draft boards.

The one group not protected were selective conscientious objectors, who supported other wars but specifically opposed the conflict in Vietnam. The hypothetical political objector in your question would probably not be given conscientious objector status for this reason. To qualify as a conscientious objector a man had to be against all wars. By the end of the Vietnam conflict many religious groups had begun to urge the government to allow selective conscientious objection, these efforts were largely halted by the creation of an all-volunteer military.

It's also worth observing that during the Vietnam war two major classifications of conscientious objectors existed. Class I-O was a objector eligible to do civilian work. Class I-O-A was an objector willing to do noncombatant military service. It was generally much easier to be given I-O-A status, as it still involved serving in the military.

A man could also avoid military service in other ways. Fleeing to Canada was one option. A legal avenue to avoid the draft was a medical deferment, which required a sympathetic doctor. Educational deferments allowed men enrolled in college or graduate education to temporarily avoid the draft.

In short, no protections existed for political objection to war. Sometimes those same convictions could be protected as long as they were labeled as "religious" by the government.