Am I able to change my Apple Mail address to a different password than my iCloud password, if it is being used as my Apple ID for iCloud, iTunes, Etc.?
I have an idea that might work. Normally, no... that would be one PW for everything. But if you turn on two-factor authentication security for your AppleID, it allows you to setup application specific passwords so third party apps like Fantastical and others can access iCloud information.
I don't see it mentioned either way in the documentation, but I'm wondering if you could create an application specific PW and use it like you described. Worth a try.