I would suggest that when you try she uses physical contraception with any other partners, and none with you. When she gets pregnant, assume it is your kid. If you really want to make sure, get a paternity test done at birth.
Since you said you want some of the kids you raise to be yours biologically but not necessarily all of them, if the kid happens to be genetically someone else's, rince and repeat. At least one of your kids should be yours biologically, and the chances of her getting pregnant from someone else when trying with you and trying to avoid with others are very low.
Other ways to limit the chances are to avoid reproductive sex with other partners while trying to get pregnant with you. So, oral sex, manual sex, anal sex, but no intercourse. That could be a middle ground that would allow for sexual satisfaction, especially if you try for a long time, and would negate the risks of getting pregnant. As to whether it's fair, I don't know, ask them what they think. It's the kind of decision that should be made with everyone involved, after all.
I can't pretend to understand why it matters to you to share genes with the children you raise, after all, I don't want any biological children, but since it's important to you, I think you should make sure to let the other people in your poly circle know right away, because it's probably important. I think another option would be to just let things happen and then take paternity tests if you and another man both want kids, because if the women who are part of the circle want several children, chances are one would be yours. But it seems to be something you really, really care about and don't want to take chances on, so...
Either way, good luck.