"To be brief, my boyfriend isn't too accepting over the topic. He says he doesn't care, but it's quite obvious by his behavior that he doesn't approve."

You and your boyfriend have a difference of opinion/beliefs. I would not lie to him; that is a lie about yourself and can not make things better, just makes you a liar. You have to decide how much it matters to you in your life. If you were to get married, you would not practice your bisexuality and you'd have to decide what that would mean to you, only. Right now that may/may not matter as you may/may not have an exclusive relationship with this boyfriend. But what he believes is his business. You don't want his religion forced on you and he doesn't want your bisexuality forced on him; I would ignore the subject or find someone else with interests more like yours in that regard.