Hi Julie,
as far as I see it there are two ways to do this:
1. LTI (Learning Tools Interoperability): This is an interface-standard that does exactly what you are trying to do: providing content from one LMS to another. In most cases that works like a SSO - the learner clicks on a button in the university LMS, the course (hosted on your client's LMS) opens in a new window. Behind the scenes, your client's LMS works as a LTI provider, the university's LMS as a LTI client. A secret key is shared and allows the two LMSs to communicate with each other.
That works quite well, but requires both LMS to have LTI functionality. Most LMS do not offer that at all, some do if you pay for it, very few offer it for free (Moodle does).
So, that might be a nice solution, but only in theory.
2. The only other possible way I can think of is to make the course visible for the public and embed it on the university's LMS. Some LMS do offer the possibility to have courses that can also be viewed if you are not a registered user of the LMS. If your client's LMS does not have this feature, you would have to export the course as HTML and host it on a webserver somewhere.
This bears the risk that users of the university's LMS find the URL to your course and share it with other people - your client may not be happy about that. And - as it is not hosted on a LMS - the university's LMS does not know if all interactions, quizzes, or whatever it is that the completion of your course relies on, have been completed.