Recording video on the browser on iPhones/iOS - this is driving me crazy

I'm attempting to access the user's webcam/phone camera to record a video and mix audio in (imagine TikTok). So the user would press record, audio would play (for the user to time their dance or whatever) and then when the audio ends, the recording stops, and they get to download it.

I can implement it just fine with MediaRecorder and CaptureStream for Chrome on Windows and Android, but no browser on iOS supports CaptureStream. Even MediaRecorder doesn't seem to work.

Does anyone know of a solution (that doesn't involve streaming the video to a server and mixing it there and downloading it back) to record video on iOS (it can certainly playback the stream on a video element) and mix it with audio?