How to get calendar invites to come in as iCal when i use a gmail address as my appleID?
I believe the solution is to force iCloud to send you invitations by email. Most people complain when they are inviting Gmail users from iCloud calendars, but I believe you're having the same problem.

Log into iCloud.com
Select Calendar
Click Gear Icon, elect
Select Advanced Tab
Select radio button for &quot;send email to [your primary iCloud email]&quot;

iCloud tries to handle invites &quot;behind the scenes&quot; when the invited email address is connected to an iCloud account. Does this work for you?