I’m not sure you can ultimately address this question without reference to faith. Or perhaps, a better term might be conviction.
A Buddhist starts with the convictions that no-self and rebirth are true. At least contingently true, awaiting our direct experience as confirmation. And if they are both true, they must be compatible.
The issue you raised has a long history of being discussed, so there are lots of ways of answering your question. If you already believe both those statements are true, you tend to gravitate towards one of those historical ways of answering the question, one that makes sense to you, and move on.
If someone questions the compatibility of those two beliefs, you respond with various answers you’ve learned, because you already believe the two positions are compatible and you are just trying to help the other person understand.
But if the other person does not come from the conviction that both no-self and rebirth are true, those answers might not be convincing. They might ultimately find the other side of the long philosophical conversation - the position that the two beliefs are contradictory - more compelling than the Buddhist response to that challenge.
In sum, you might not get an answer to your question that satisfies you. But observe that we all seem quite satisfied with the answers we are providing.