Customizing the CLASSES assigned to the TinyMCE editor wrapper

I’m creating a site in WordPress that is piggybacking on an existing institutional stylesheet. Many of the stylesheet elements take on a different behavior when they are in the container class in which normal body text resides on the page. This class is called “webid-container”. So the stylesheet is full of things like

.webid-container h1 { ... }
.webid-container b { ... }

So of course I want to have the content within the editor look like it does when it is inside a .webid-container classed wrapper.

However I cannot find a way to specify a class to assign to the editable area of the TinyMCE editor which would accomplish this.

Of course I could search for every place in this massive institutional stylesheet where “.webid-container” exists, and copy it to my own local stylesheet and replace all instances of “.webid-container” with “.wp-editor” and it would probably work, for now. But as soon as the institutional stylesheet would change, I’d have to keep my local copy up to date.

It would all be so simple if I could somehow tell TinyMCE that I want to add an additional class alongside “wp-editor”.

Any ideas?