Hiya Duck:
You really screwed up now... AMEL is a BIG addiction! Once you get 2 throttles in your paws (webs?

), a single just won't cut it anymore (unless it’s on floats, but that's ANOTHER addiction!

)
The Duchess is so benign on 1 engine that you should be able to hold the rudder easily- barring any physical infirmities.
Once you go through the dead engine recognition drill, then you can set trim if need be; just remember it will give you grief when add power back to the "dead" engine.
You never have to really hurry with the engine-out drill, but you do want to feather the dead prop before RPM falls below 1000 or the stop pins will pop out preventing it. You should have somewhere between 45-60 seconds to do the whole drill all the way to feathering and that is a lifetime!
Once the dead engine is secured, take a breath, relax and then go thru the checklist for single engine operations (cross-feed, trim, speeds, restart, etc). You'll be surprised to note that the BE-76 can hold altitude just fine with you and the MEI in it.
Just keep the plane at Vyse ("blue line") or above and the bird will fly just fine.
One major thing though: If its hot or high, the plane WILL NOT climb worth a damn, so plan on aborting take-offs straight ahead or stopping on the runway. If you get airborne to an altitude that won't let you run in to anything, REGARDLESS HOW HIGH, level off, and go through the drill to ID and feather and then LAND!
If the engine fails in the VFR pattern, or on the approach (an ILS, I hope!), declare it and just fly normally- don't do anything except identify the dead engine, feather the prop and secure the mixture on the DEAD engine (You wouldn't believe the number of times the GOOD engine mixture is pulled!).
Leave the rudder trim alone; as you decelerate for landing, the plane will come about naturally.
Don't worry about doing any cross checks or restart drill while in the pattern or on the approach- they're pointless as you are going to LAND and sort things out on the GROUND where you can't get hurt!
You DO want to tell the examiner and your MEI your plan, so they won't think you're a ninny. ;a good place to do that is during the pre-flight, take-off and landing briefs.
One final thing: Don't forget to do the GUMPS check; it's way too easy to forget it in the heat of the moment.
By the way- all this comes from personal experience. I've had more than a few actual failures, 1 of which was in a Duchess, just as I got the gear up!