I don't mind the douchey, bitter Trevor. He SHOULD be this way when you and your family are HATED by all.
Keep in mind, the Belmonts in this period of time aren't reviled as the famed, monster-hunters they are. They're hated, ostracized, and considered deplorable outcasts. A once noble house regarded with disdain and skepticism because they're not understood/misinterpreted, so no wonder Trevor is a bit of a loner, alcoholic, and spiteful towards everyone...everyone has heard about rumors about his clan and believes him! They see him as not much better than Dracula's creatures. Even the Church fears the Belmonts (as the Bishop hints at them being excommunicated) and has likely helped propagate their negative image.
But you see that despite it all, he is the reluctant hero, and the Belmont clan has continued their tradition for being deadly vampire/monster killers who at the end of the road, are moral and care about justice after all. Trevor even comments that he helps the Speakers (Sypha's clan) because he feels empathy towards them for being shunned by the masses who are under sway of the church. And he comes around to helping rally the villagers against Dracula's creatures and the Church that was misleading them.
In Castlevania lore, the Belmont family ONLY becomes respected and infamous again because Trevor (and his allies) defeat Dracula. By the time Simon Belmont takes up the Vampire Killer, the Belmont clan is famous again. I hope the series continues so we see Simon as well! Different seasons based off the different games would be awesome!