I found a simple link that states in pretty clear language what the COPR is. In most respects, it is basically an updated record of landing, except that unlike the record of landing, it's not intended to be used as ID on an ongoing basis.
https://www.canada.ca/en/immigratio...ry-confirmation-permanent-residence-copr.html
Now the comments above are technically correct that a COPR does not show that one IS a PR on any arbitrary date - because technically PR status could have been revoked afterwards. I say 'technically' because that is also true for someone bearing a valid PR card, and the PR status could - technically - have been revoked any day after it was issued.
In my view, the difference is - therefore - that with a PR card, it's simply
presumed that the valid PR card is recent enough that the bearer is still a PR,
and that the permanent resident was (relatively recently) still in compliance wiht the residency obligation (and therefore relatively low risk of being deemed inadmissible).
Now, I don't know if this helps you. The people you're dealing with clearly don't have entirely up to date information, nor do they know how to find out exactly what they're looking for, simple confirmation you are still a PR.
[We could have a side note or argument that it remains, in my view, a major gap in what the government does/can provide for PRs - confirmation when needed they are still "PRs in good standing." But, to use the cliché, it is what it is, and all you can do is try.]
So I think some of the comments in this thread miss the point: what we know usually to be true, what (from that) the military 'should' require, what they actually require (when they don't actually know), and what you can get and provide are all distinct things.
I'd summarize this way:
1) COPR actually IS a reasonable 'identification' document or proof that you were - as of the date you landed - a PR. Relatively few PRs lose that status.
2) The expired PR card is further evidence that you were a PR - and again, not so many lose that status.
3) You have copies of your application for a new PR card, along with the evidence - provided with that app - that you have been residing with your spouse abroad, AS PERMITTED BY and provided for by the immigration law. [I understood your spouse is a CAF member serving abroad, and if so, that should serve as presumptive evidence - if they know you're living together abroad - that you are also in compliance with the residency obligation.
[It's actually a bit strong than that since time abroad with a spouse who is a crown servant - including military - also counts toward citizenship. This is NOT a status that is open to question, generally:
https://www.canada.ca/en/immigratio...citizen-permanent-resident-living-abroad.html ].
4) You can also show them references to government documents and manuals and the immigration law, which probably won't convince them to decide (they're not immgiration specialists), but might convince them to keep the file 'alive' for the time being.
The reality is - based on my experience with bureaucrats - you're probably going to keep putting forward the evidence you can in the hope that they'll not cancel the file/your application and both you and the hiring people will hope that the PR card or something else comes through to let them do what they want. Maybe they'll find some way to let you do this work or some related work on some nebulous temporary basis. Maybe they'll find someone senior who can override (temporarily) the process for a bit, or 'pending receipt of PR card.' Maybe some confirmation will come quicker than you expect, or somebody will decide the acknowledgement of receipt or some other doc will be enough for a bit.
Problem is, of course, we have no idea how long the PR card might actually take.
Good luck.