First of all - answer or don't, but I'm curious. How old are you two? I'm guessing very early 20's if that, probably 17-20. NOT KNOCKING you in any way, whether I"m right or wrong. It's just that the issues and the questions seem... Inexperienced. It 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 you, lol. Most of my relationships have been with bi or (formerly) lesbian women. I have ZERO issue with them having girls on the side, be it "just a fuck" or a full-blown relationship or anything in between. Just doesn't phase me, no jealousy, nothing but "have fun sweety". :shrug: The fact that most of these girls were not onay 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, and Ceoli can chew me out for it, 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 relationship, 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 ou are, blah blah - but it matters to me A LOT. So - to relate this to the OP's situation...
Well, you're straight. She's bi. though ome 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 dalliences that they do not with whichever gender. Clear as mud, hope ya'll got it, lol.
On to 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, lol. If she feels that you being physically involved is not okay because it's cheating, than she shouldn't be bringing you in to the situation. There are instances where this would "make sense" or "be okay" - but on a level of understanding each others needs, desires, and drives that I am decidecly NOT sensing is there with you two yet.
*sigh* I'm getting long. I'll try to clarify later, lol.