200 meters from Sant’Ambrogio doesn’t tell me much. 200 meters is a long distance which can put you in a great spot or not so great spot, or on a quiet spot or a noisy one. What is the exact address? Piazza del Carmine used to be noisy at night when I was young in the 1980s because of the presence of the famous Bar Caffè La Dolce Vita, a super trendy place that attracted a lot of youth in the Piazza. The bar has closed down since, so now the place is certainly quieter than those years.
In. terms of distance from the center, both are in the historical center, but both are about 15-20 min walk from the Duomo or Piazza Signoria. So if we take Piazza Duomo-Piazza Signoria as indicative of the most central center of the city center, then the two locations are probably equidistant. There is no need to walk across Ponte Vecchio to go from Piazza del Carmine to Piazza del Duomo or Piazza Signoria. Both Ponte Alla Carraia and Ponte Santa Trinita would provide a shorter walking path, although I don’t see anything wrong with walking across Ponte Vecchio.
In terms of neighborhoods and foods, both have a plethora of great options. the district of San Frediano, in the Oltrarno (Piazza del Carmine) has become trendy in recent years, so some of the best restaurants and the coolest bars are certainly there (if you want I can provide some names).
If the choice were on the neighborhood alone, San Frediano would be my choice, but again it depends on your Airbnb/hotel accommodations as well.
A/C is generally not needed in October, as days are shorter and at least during the night temperatures cool down significantly. During the day it may still be warm, but nothing compared to the scorching months of July or August.