Wonder why the school you did your flying at, didn't tell you. Probably didn't want to sell you the SE time when you could fly twin. Also, you'd have to consider the extra proficiency training for VFR maneuvres for the checkride and therefore also rental and DPE fee on top of all the other stuff in twin. I'd not do only one class CPL in fixed wing. I know it sounds silly to some, to have CPL in twin, but not legal to fly single - normally the common sense dictates that twin privileges - more complex/advanced, would mean SE built in.
It was 2 weeks of jumping hoops, and was rushed, as well as the instructors having other stuff to focus on, we were just ticking boxes as best we could. The flight school was actually an offshoot of the company I was working for elsewhere so was a 2 week stop in the US to get all certified etc then off elsewhere to fly N reg aircraft. No cost incurred by me thankfully. Never needed the SE as all the work was multi engine anyway. Weird system as you say with the added complexity of twin ops, but you can't fly a single.