Your system conditions seem pretty violent,
i dont think you can have a s type orbit , becose once you leave the orbit of the first star your planet is continuing a inertial movement beyng atracted from the second star ..
how to explain.. imagine trying to spin a rope with a weyght at one end , once you let it go , it wil stray away , and other person has the imposible task of catching it,
so an s-type ofbit is kind of unstable orbit,
in your context your exomoon is orbiting its gaseos planet and influenced by your double star :>> seasons i dont think you cant have becouse violent conditions of the gravity exerted form your dual star system and your gaseous star influence ,
your storry must be altetred a bit, consider the double star is in the center of the system conected in a gravitational push with each other
your gaseos planet has a eliptical orbit at a distance of 20 diametres of its first star.
and now your gaseos planet is further away, and your exomoon isnt much influenced from the double stars form center of the system , so then i think that stil the conditions are to be less harsh on the moon's surface and waters so your only inhabitable options are more forgivable. still i think your living options should be on the surface and under.
NOW ! you can decide how many seasons you can have 2 or 4 ..
please feed back me..