"We apologize for any inconvenience" is a set phrase, so anything outside of the four or five variations that are commonly used will sound unnatural to natives. "We apologize for the inconvenience" "We apologize for any inconvenience" "I apologize for any inconveniences occurred" does not seem correct to me, and has almost no entries in google. I would use one of: "I apologize for any inconveniences that have occurred" "I apologize for any inconveniences incurred" That last one is actually quite clever, as it is blisteringly sarcastic if the speaker is not in the same group as the listener.

It's also common to just say 'We apologise for any inconvenience'.

Yes, It's natural and correct! You can also say "any inconvenience caused"