Should I suggest to my girlfriend............. (bisexual male)?

if you really want to suggest this to her then you would just have to suggest it when the time is right and when you are in the mood, however you shouldn't feel like you should have to do this, after all, the real problem is that she assumes that bisexuality is where you want intimacy with 'both' genders, when in reality bisexuality means the possibility of 'either'.