In the USA at least legal 135 ops in twins require loading so that you have single engine climb. Part 91, not so much.
Australian regulations IFR charter AND IFR airwork aircraft (below 5700kg) must be able to climb at 1% gradient to 5000ft in ISA (not ambient) on one engine (critical engine failed and feathered, clean etc). For VFR and all private operations just the ability to maintain any height up to 5000ft in ISA is required on one engine.
An airwork (training) PA-44 for instance to comply with the IFR 1% rule needs to be loaded about 2 POB and 2/3 - 3/4 fuel. For VFR operations it can achieve MTOW. I can say that under these rules the aircraft easily flew on one engine at blue line and climbed if flown correctly. I've also flown PA-31 at high weight with the engine feathered and it performed more or less to book, climbed and was able to circle to land. If your twin is not able to climb on one engine and the book says it should, I would be asking your engineers some serious questions. Not saying you should go out there and pull engines to test this, get an experienced instructor and play around at a safe altitude over safe terrain, as there is also the chance of other things going wrong.
I'll add that there is also the requirement that all aircraft must be able to climb at 6% gradient after take-off on the day with the gear down.