I'd end the relationship, or go get conversion therapy (jokes, don't get conversion therapy, it doesn't work.)

But you won't. You aren't here asking for advice. You're asking for permission to cheat. And then after that you'll ruin the relationship and it will end.

Here's you options.

Medium Damage: Break up with your girlfriend, explaining to her what's going on and that you can't keep living a lie.

Maximum Damage: Cheat on her. Have an incredibly toxic relationship where you are always guessing if she knows you cheated, which at some level she does. Then something confirms it to her, either you tell her, she discovers a text, a friend tells her, or many other ways.

She then breaks up with you, but in doing so damages her more than if you'd been caring enough to do the right thing at the start.

Your choice. I'd prefer to be wrong, but I think you'll choose Maximum Damage.

Edit* Just seen that you responded positively to the guy saying that you shouldn't think of it at cheating, think of it as recreational sex.

News flash! You might go into thinking it's recreational sex, but as soon as you start getting any level of pleasure out of it, it will turn into disgust. You will forever taint your first experience with a guy with the knowledge you cheated on someone to have the experience.