John Tyler, oddly enough
He had actually served as chief executive (granted, in fairly atypical circumstances, but the rebellion was atypical as well); he had co-chaired the 1861 peace conference, and so - perhaps - might have been able to negotiate some sort of short of war situation; he had experience dealing with the British on a nation-to-nation level; etc.
Of course, he was a Virginian, and so presumably would not have been "available" at the same point Davis was in 1861; and he was elderly and in very poor health, dying in January, 1862.
Nonetheless, of the antebellum former presidents still alive in 1861, he was the only southerner...
Best,