First of all, answer or don't, but I'm curious-- how old are you two? I'm guessing very early 20s, if that, probably 17-20. I'm not knocking you in any way, whether I'm right or wrong. It's just that the issues and the questions seem... inexperienced. This matters, in that some advice will be different based on age and experience, and rightly so.
Getting away from that, I think my perspective is a very interesting blend of the two of yours. Most of my relationships have been with bisexual or (formerly) lesbian women. I have ZERO issues with them having girls on the side, be it "just a fuck," or a full-blown relationship, or anything in between. It just doesn't faze me. I have no jealousy, nothing but "have fun, sweety."

The fact that most of these girls were not okay with me having other girls made perfect sense to me. I was less than thrilled about them having other guys. Girls weren't competition. So I get where she's coming from there. (Ceoli can chew me out for this. lol)
I have recently had to bend my brain around the idea that other men can be okay too, not because it's come up in the relationships, but because Violet lets me have other women. And though it took a lot of convincing for me to decide it really was okay, since that's been acted on, more than once, I have to come to grips with the issue in my head. Namely, if she DID decide she wanted another guy, but wanted to keep me around, I feel that it should be okay, lest I be a hypocrite (worst sin possible, in my book).
I know, I know-- don't change who you are, blah blah-- but it matters to me, A LOT.
So, to relate this to the OP's situation-- You're straight. She's bi. Though some here say that for a truly bisexual person there should be no difference, I disagree. Bisexuals get twice the theoretical action, by default. And I do NOT believe that because they get to have a dedicated relationship with one gender while having involvement with the other as well, that they should "have to" allow a "straight" partner dalliances that they do not with whichever gender. Clear as mud? I hope ya'll got it. lol
On to the other issues, of which there seem to be many. If she feels it's cheating, despite your assurances, than she needs to not do it. If she feels that you being physically involved is not okay because it's cheating, than she shouldn't be bringing you into the situation. There are instances where this would "make sense" or "be okay," but there's a level of understanding each other's needs, desires, and drives that I am decidedly NOT sensing is there with you two yet.
*sigh* This is getting long. I'll try to clarify later.