Q:

There should be a way to manually invite users without enough rep to chat

I regularly see postings from new users that don't quite fit the criteria of the relevant site.  In many cases, the posts are commented on and downvoted/closed, but the user simply never comes back to respond.  But in other cases, the user is obviously invested in their question, but doesn't agree with or understand the position of the community regarding their post.  Here's a recent example.
Since argumentative problematic posts get so much attention from regular high-rep users, they tend to generate a lot of downvotes as well; this harms any chance the new asker has of gaining the 20 rep required to join the chat and have a more personal discussion.
In instances where a new user isn't fully understanding why their question is inappropriate, I feel like it would be nice if someone (a site moderator perhaps) had the ability to manually permit them to participate in chat temporarily even if they don't have the rep required.

A:

This already exists. Room owners can go to the room's access tab and add users to the explicit write access section:

