I know I lose my next turn, but is that ability still available to me?
A: You can only win with the Secret Cultist if you are playing it as an out-of-turn play. If you are forced to play it as an actual in-turn play, you take the penalty instead of the out-of-play ability. Thematically, the idea is that you’ve been exposed as the Secret Cultist and your plans therefore ruined. They locked you up until you came to your senses (your lost turn) and you no longer worship the Ancient Ones.
Note that if you play the Secret Cultist as a counter-Surprise, it does not count as using its in-turn function, and does not “reveal” you, so you don’t take the penalty. It’s more like you were a Secret Cultist for a while, but then you got tired of hanging out with those losers, and just decided to leave on your own.
Either way, whether you’re thinking about it thematically or not, there’s no way you could retain that special status/ability. Since you can only claim your Secret Cultist win by playing it out-of-turn after the game ends with an UnGoal, you need to still have it in your hand when that happens. If you’ve already played it in any fashion, that card has been used up, and is now in the discard pile. You’re no longer the Secret Cultist. In fact, if the discard pile is reshuffled, someone else might end up with it in their hand, in which case THEY would be the (as yet unrevealed) Secret Cultist. You’re just some failed has-been wannabe Secret Cultist (or successful survivor of a dangerous cult experience!)