From Franklin's writings, we do know a few certain things about his creed. He was a theist, that is he believed in an active personal, as opposed to a cold and distant God. After a long debate in this comment thread, I am convinced that Franklin rejected the orthodox Protestant doctrine of sola fide (or that men are justified through "faith alone").
Yet, Franklin was also skeptical that men could save themselves through their good works alone. His letter to George Whitefield demonstrates he didn't think HE HIMSELF could do it. Though, he may have remained open to the notion that some could.
That said, Franklin's writings demonstrate he thought good works and virtue central to the salvation scheme. Good works were a necessary component for salvation, but also, in the end, insufficient to merit eternal bliss. Some Supreme Act of Providential Benevolence would come in and save the day for most, if not all folks.
If Franklin believed in Christ's Atonement, it was in this (unorthodox) sense. It was more than just choosing between a "Universal" versus a "Limited" Atonement. But rather a Universal Atonement that borders on, if not results in the eventual salvation of all men's souls.
I see it as a "plus factor" Atonement. That benevolent, divine push that expedited salvation for all good men. Good men, because of their virtue, and regardless of their exact faith, were first in line for Heaven. If there is a scholarly term for this atonement theory, I'm not aware of it.
Belief in Jesus was important for salvation not because men had to put their faith in Jesus' finished work on the cross, but rather because Jesus perfected morality. If morality was central to salvation as Franklin believed, and if Jesus was the perfect moral teacher, then it stands to reason that Jesus' followers, whether Trinitarian, Arian, Socinian or a believer in some other Jesus centered system, would be closer to the front of the salvation line, IF THEY SINCERELY FOLLOWED HIS MORAL TEACHINGS.
As it were, I don't think Franklin had too much concern for the souls of the Arian Richard Price and the Socinian Joseph Priestley. As he noted to B. Vaughan, Oct. 24, 1788:
Remember me affectionately to good Dr. Price and to the honest heretic Dr. Priestly. I do not call him honest by way of distinction; for I think all the heretics I lave known have been virtuous men. They have the virtue of fortitude or they would not venture to own their heresy; and they cannot afford to be deficient in any of the other virtues, as that would give advantage to their many enemies; and they have not, like orthodox sinners, such a number of friends to excuse or justify them. Do not, however, mistake me. It is not to my good friend's heresy that I impute his honesty. On the contrary, 'tis his honesty that has brought upon him the character of heretic. I am ever, my dear friend, yours sincerely, B. Franklin.Drs. Priestley and Price were honest, virtuous men and that's what mattered most, in the grand scheme of things.