I wouldn't ask him to convert or to be baptized. I would ask that he attend inquirers classes to learn about Orthodoxy. Being "open" to raising the children Orthodox is not a guarantee. He needs to know what Orthodoxy is so he can know whether he would allow your children to be raised Orthodox. Until he is informed about the faith, you shouldn't proceed to much farther into making the relationship more "serious."
All of this is a moot point, since AFAIK, Orthodox Christians cannot marry persons who have not received a Trinitarian baptism, in the Church. If an Orthodox Christian chooses to do so, then they are also not considered "in good standing."
Unfortunately, there is no perfect answer in these situations, that does not involve the potential for misunderstanding and hurt feelings.
When we make choices or decisions that are not what the Church believes, preaches and teaches, such as deciding to marry a non-Christian, then by our choices, we have voluntarily placed ourselves outside the community, outside the unity.
As you know, the Orthodox Church believes that marriage is more than a private transaction between two people. Rather marriage is a sacrament, an event in which Jesus Christ Himself participates. If we understand the "ecclesial" dimension of marriage, then we also understand why marrying a non-Christian is a personal choice that places us outside the Church. People are never forced to marry “outside the Church” – that is the decision of the individual. Orthodox Christians who choose to enter into marriage with a non-Christian are no longer considered in good standing with their Church and are unable to fully and actively participate in its life, including receiving Holy Communion. This self-imposed restriction on full participation in the sacramental life of the Church is a matter not only of ecclesiology but also common sense – if we do not believe or follow the teachings and practices of the faith, why would we want to participate in its Sacraments?
I encourage you to speak to your priest and discuss your feelings with him. He will explain the matter much better than I can.
I know that you have shared elsewhere your feelings on being raised in home with parents who had differing beliefs, although both were Christian, and how this resulted in feelings of not belonging anywhere. To a certain extent, I understand these feelings since my parents were of different beliefts as well. Although they were loving and respectful of each other's beliefs, I'm sure it was very lonely for my mother to go to church alone all those years, and vice versa. As a small child, I couldn't understand why I would go to one church with Dad and another with Mom, and they were never together. Children notice these things and I felt lonely and afraid.
It's not something I would want my child to go through. Nor would I want my faith, the deepest, truest part of my self, to be a source of conflict or compromise.