For future searchers - I was able to get this working. The missing piece was sharing the cookie between the native side and the webview side. I thought this was already happening, but it was not. This ended up being simpler to fix than I expected - since I’m using react-native-webview, just setting sharedCookiesEnabled = true allows the persistent cookie that’s set during the native auth flow to get passed to the webview and everything proceeds swimmingly from there.
Cheers!