Ayumi H. answered 04/16/19
Native Japanese Speaker with teaching and business experiences
我々が 起こした偉大な犯罪の謝罪 (われわれが おこした いだいな はんざいの しゃざい)
You are right. About this line. This is not a complete sentence. It looks like a headline. If it is an apology in a complete sentence without changing so much wording, I would write: 我々が 起こした過大な罪の謝罪 (われわれが おこした罪の しゃざい)いたします。
As you pointed out 偉大な should not be used In this case. The word is often used to compliment achievements and successes. Also, I would not use the word 犯罪 in this situation. It feels a bit too much for the situation like this ( assuming I have all the information to judge the choice of words)If it was up to me, I would use 過ち (あやまち) instead. There are many ways to apologize in Japanese and I would prefer to use a different expression such as お詫び申し上げます。or 深くお詫び申し上げます。
我々 sounds too formal and less friendly as well. I would change to 私共 and rewrite as below:
この度の私共の多大なる過ちを深くお詫び申し上げます。(このたびの わたくしどもの 多大なるあやまちを ふかくおわびもうしあげます)