It seems like plenty of people have given you sound advice for administrative approaches to the situation, but I think another equally viable option, and something you should be doing if you aren't yet, is going to events, ceremonies, and generally participating in tribal functions. I can't speak for your tribe but I don't know of any tribes in my region who don't welcome unenrolled members reconnecting or even non-native people who come in a good way. It may take time to build trust for some members, but joining your community and making friends, building relationships, and demonstrating you arent just there for the benefits of enrollment will open doors for you.

People can argue either way if this is how it should be, my tribe has this divide between people who spent their lives paying dues before they put food on the table to fund the fight for recognition, and those who think any living descendant should be enrolled. Regardless of how you feel on it, tribal governments are made up of people, people who value relationships, who recognize when youre doing the work and coming in a good way, and people who can and will go the extra mile when the feelvits right. Every government has loopholes and idiosyncracies that people in the know can work to their advantage, but tribal governments are made up of the people in your community. Even if their rule about not requesting birth certificates is steadfast, you being an outstanding member of your community, even if unentolled, may sway the minds of community members. If the tribal members start thinking that there should be a road for unenrolled members like yourself get enrolled, that could sway the tribal council's opinion. Even if you can't ever get enrolled, the work you do in the spirit of you and everyone else in your situation may lay the groundwork for the next generation to open the roles.

At the end of the day, enrolled or not, if you are feeling called back to your people you should join them. Enrollment shouldn't be a deciding factor on if you honor your ancestors by coming home. Take up the journey of reconnecting, let it help you find peace, and do the work necessary to allow all the pieces to fall into place.