This is a nice idea! I’ve often wanted to be able to do this as well. I think all the info you need exists in wordpress, except maybe the notifications and stuff that live updates as things happen in discourse. I think you could get pretty close by just copying the html and css to replicate what you see on discourse on wordpress. Would be nice to have a recipe for this here so if you get there, please share your results!
In case you are curious, the Legal Empowerment Network took a different approach. Feel free to join to see how it feels as a user. The utility bar across the top looks different and replicates the usual wordpress way of doing things. It also provides access to some additional features that exist only in wordpress and not in discourse. The wordpress site also serves additional purposes that are not directly about the network, so keeping them a bit separate works towards that goal too.
This is a little unsatisfying because the experience is not seamless. Would be preferable to have one utility bar that is the same across both platforms.
The way most sites do this (you probably know this too) is to put a header across the top of both wordpress and discourse that is the same, with the discourse header below that.