Q: iCloud-Gmail calendar conflict

My Apple ID and iCloud are registered to my Gmail account. I use both in my calendar and this is causing conflict when I receive calendar invites to my iCloud account. Rather than having the invites pop-up in the Calendar program on my Mac, I receive it to my iCloud email address - when I click approve (though having to open my iCloud email is cumbersome since I don't use it for anything) it takes me to Gmail and gives an error.

When I get an invite to my Gmail account, I can't do anything with it but accept or discard the invite (can't for example move it between the different calendars I have in Gmail or from Gmail to iCloud).

I tried changing my iCloud/Apple ID email to the iCloud one, though it tell me I can't use Apple domain for it..

Essentially what I want to achieve is to get my invites to my calendar program when I'm invited using iCloud (which is associated with my Gmail). It is driving me crazy and can't seem to find a solution.

Anyone knows how to solve this?

Cheers,