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.

The "correct" Apple ID is the one used to purchase the apps originally.

It doesn't need to be tied to your current email address & in fact doesn't actually have to be an email address at all. You can change your primary email address associated with an Apple ID...

Change your Apple ID

Follow these steps to change the email address that you use as your Apple ID.

1. Sign out of every Apple service and device that uses your Apple ID.
2. Sign in to your Apple ID account page.
3. In the Account section, click Edit. Under your Apple ID, click Change Email Address.
4. Enter the email address you want to use as your Apple ID. A verification code will be sent to your new email address.
5. Enter the verification code from the email into the fields provided and start using your updated Apple ID. Didn't get the email?
6. Sign back in to all Apple services with your new Apple ID.

You can change your Apple ID to any other email address you control, as long as it's not already in use as an Apple ID. If your email address ends with @icloud.com, @me.com, or @mac.com, you might not be able to change your Apple ID. You also won't see Change Email Address on your account page.