Just a note which I think can be useful. 约会 usually would imply a date, although the basic meaning is just a meeting, usually casual, such as between friends (as is in the present sentence, which has no problem). For a business meeting or appointment for service, especially the latter, it seems a little strange to say 约会. An alternative for such cases is 預約.

Is there a reason "对不起" is not accepted for "Sorry"?

"对不起，我有一个约会" still not accepted, 13 August 2018.

对不起我有一个约会. Still not accepted. January 9, 2019

"一個" is unnecessary. It's too literally translated.

不好意思，我有约了

对不起 should also be accepted...

Is the 一个 needed? Would 不好意思我有约会 also be okay?

You don't need "一" here.