Take a copy of your executed marriage licence application - after everyone has signed it. Submit a copy of that, along with a fee paid receipt (for the PR application fees) AND another application to extend his visitor status. Explain to them he'd like to stay together with you while the application is in process.
If you want to guarantee he can remain, file a "spouse-in-Canada" class application ("Inland"). That means he won't be able to leave Canada at least until he receives his open work permit (and it's not recommended until PR is granted). But if he includes an application for an "open work permit" with the PR application he will gain "implied status" and will not need to leave Canada.
If he wants to go home, then by all means get the medical done in Canada, it's likely to be less expensive and more convenient. Sadly, it is unlikely he will be able to obtain a TRV to return to Canada, so he should be prepared to wait out the application process in Zambia. But if you wish to go this route, you should apply for the "spouse-outside-Canada" application ("Outland"). He does have the option of applying via the Outland route even if he is in Canada, but if they require an interview, it will be in his home country generally - and then it is unlikely he can obtain a TRV to return to Canada.
Good luck! Don't panic, you're in good shape.
