Cannot see/accept/decline invitations from Exchange
This is not a great answer, but it's a workaround for now. I set up an account on Outlook.com. Invitations to or from that address work seamlessly, so both my wife and I can open each other's invitations, accept, decline, etc.

The downside is that this only works for events on my Outlook.com calendar. There's still no way to accept an invitation from her and put it on my iCloud calendar, and she still gets an attachment if I send an invitation from iCloud or some other account. So, I very much consider this a workaround, not a solution, and would appreciate any actual answers.