In 1963, The Doomsday Clock was moved 5 minutes back, to 12 minutes to midnight, as opposed to 7 in 1960. How did the committee at Bulletin of the Atomic Scientists justify this in light of the recent Cuban Missile Crisis? Was there a perceived "cooling off" period immediately following the crisis?

Your question is in two parts - on the first part:

"How did the committee at Bulletin of the Atomic Scientists justify this in light of the recent Cuban Missile Crisis?"

Firstly, it is important to note that the committee did not decide the time of the Doomsday Clock. They only did so from 1973, when the Bulletin Editor, Eugene Rabinowitch, died. From its inception to his death, Rabinowitch decided if the time on the Doomsday Clock should be changed. So we are not talking about a committee, but about the opinion of a single person. As a nuclear physicist (also co-author of the Franck Report), he was better qualified than most to decide, but we are still talking about the decision of a single person.

This appears to be a fairly common sort of question because on the FAQ section of their website, one question is:

"Were the hands moved during the Cuban Missile Crisis in 1962?"

You can view the question and answer here.

The committee's retrospective justifications are as follows:

"Too little was known at this time about the circumstances of the standoff or what the outcome would be".
Following the crisis, the US and Soviet leaders established a hotline between Washington and Moscow.
Nearly a year after the crisis, in August 1963, the two powers signed the Partial Test Ban Treaty. This, to the committee, appeared to be a substantial change towards addressing their concerns over the proliferation of nuclear weapons.

One mistake they have made: US and Soviet leaders "signed the Partial Test Ban Treaty outlawing underground nuclear weapons testing—the first treaty addressing the nuclear weapons threat". This is incorrect. The Partial Test Ban Treaty banned the detonation of all test nuclear weapons except those detonated underground.

Secondly, it is clear that Rabinowitch knew fully well about the dangers posed by the crisis. In the 1963 Volume 19 Bulletin of Atomic Scientists, Issue 1, Rabinowitch writes the opening with a short piece entitled "New Years Thoughts". From this:

"In 1962, mankind came as close as never before to the abyss of a nuclear war". You can view this here [may require access].

Rabinowitch was therefore, unsurprisingly, fully aware of the danger posed by the crisis.

It appears that what was most significant was the signing of the Partial Test Ban Treaty, which "represents progress in at least slowing the arms race. It also signals awareness among the Soviets and United States that they need to work together to prevent nuclear annihilation". You can view this here.

I hope this helps in understanding why Rabinowitch saw the change as justified.

Sources:

The committee's website.
On the life of Rabinowitch: Long, Franklin: 'Eugene Rabinowitch: An Appreciation', Bulletin of the American Academy of Arts and Sciences, Vol. 26, No. 1, (Oct., 1972), p.12-18.
Rabinowtich, Eugene: 'New Year's Thoughts 1963', Bulletin of the Atomic Scientists, Vol. 19, (1963), p. 2.