I would pick late spring, well before hurricane
season but hopefully after the risk of a cold front.
US - Bermuda - Azores - Gibraltar
is nice and would break the trip up into nice parts
, but you do run a high risk of hitting the Bermuda High with calms or contrary winds going this route. Leaving Bermuda almost certainly need to swing north to catch the westerlies.
If you can access GRIB files, like passageweather.com I would try to plan the route to go just far enough north to stay out of the Bermuda High. Hang somewhere from N FL to Charleston until you get a good window and westerlies to Bermuda and go. In Bermuda check the location of the high and how far north you have to go to stay out of the calm zone to plan the route to the Azores.