However, what I'm not willing to suspend is that the Bat is capable of reaching the minimum safe distance of six miles off the coast of Gotham
A 4 MEGATON bomb wouldn't have a blast radius of 6 miles. "Little boy" was 15 KILOTONS and had a blast radius of 2 miles.
I'm thinking at least 30 miles blast radius and therefore Batman and Gotham would have been doomed.
A 4-megaton bomb has a blast radius of roughly 13 miles.
I played around with Nukemap. Assuming Gotham is near the site of Atlantic City, you get this
. Assuming New York, you get this
. Both maps are drawn assuming a water detonation that doesn't touch land.