The player isn't given the option to refuse because that option would result in the MC quitting the Club before the first chapter - i.e. beyond the scope of the game. Being used as a sexual prop is precisely what he signed up for.
That is simply not true.
Every discussion about the job was about how he would manage the girls and convince them to keep working there. There was nothing at all about him doing any sex work himself. That conversation with Kathleen where she suddenly demands that we whip out our dick because we had previously agreed to it was terrible writing because we never once agreed to anything of the sort.
Before that scene, the player's character was holding almost all the cards. The club was desperate for new help to replace the worker that quit just before their most lucrative time of the year. The player was very reluctant to join because he didn't want to risk his future career by getting involved in any illegal activities, which was clear to August who said that "I can see in your face that you're already half way out the door". They were throwing money at him out of desperation because that was their only choice.
We also need to recognize that the player character is also very close to being a nepotism hire. He should be getting preferential treatment from Kathleen, instead of her injecting him with experimental date rape drugs and everything else she's doing. There is no good reason for him not to call her bluff and tell her that he isn't there to whore himself out. They need him more than he needs them. And he should be smart enough to see that if they are going to suddenly change the rules like that, he should quit as soon as possible before they demand even more.
This is the single biggest fault with the game. It starts off as a set up for a game where the player can either gladly descend into depravity, or carefully try to walk the very thin line where he keeps his hands as clean as possible. Then it just suddenly turns on a dime and say nah he's just a desperate whore who agreed to all this.
But not always.
There is one scene later on in the game where the character can refuse Kathleen, which is when she demands that he be filmed for a porno and he can instead have his friend do it instead. This was what I thought the whole game would be. The choice to either do the dirty work yourself, or pass it on to Killian instead. That scene also made use of the character traits we chose back at the beginning of the game, which is also commendable.
Unfortunately, the refusal does comes a little too late, as the MC already knows that the place has security cameras everywhere, so he's already certainly been filmed before. But the scene is still a good example of what the game could've been if the developers had fully committed to reaching it's fullest potential.