My apologies - I missed the specific requirement for custom properties. Unfortunately it's currently not possible to add custom properties in this way via the REST API. I have raised this internally and such functionality is being looked at, though I can't say if or when this will be made available.
Sorry for any confusion on this, but in fact you
can add custom properties via REST API for your stream target. It is not currently documented (as we've both discovered - I'll look at getting that amended) and the method is to use extraOptions, for example: