The nose dropping when inverted has nothing to do with dihedral. The strong tendancy to roll out from inverted is, but that wasn't the question. Flying a hi-dihedral plane inverted does take more attention on the aileron stick to keep it level, but that doesn't affect the elevator.

The plane having a flat bottomed airfoil will increase the amount of down elevator necessary to keep the nose up. So, if you have a radio that has fancy end point adjustments, you can set it up to give more throw down than up. But I'd do this after other adjustments.

Most likely, the CG is too far forward. Move your CG back about a 1/4" and give it a try. Be careful, but most trainers have a very broad "safe" CG range, and the instructions usually put the CG way at the front of that range. Some trainers can fly with the CG back as far as 2" from what is given by the instructions. So, if you see a slight improvement, and the plane isn't too twitchy, move it back another 1/4", repeat as necessary, but keep in mind that the elevator will become more sensitive, and there is a point where the plane will become very very hard to control.

The incidence I was talking about was, as mentioned, the relitive angles of the engine thrust line, wing, and horizontal tail. I'm taking all measurements from a referance point of the fuse center line here. If the wing is at, say, a positive 2degrees, and the stab is at a negitive 1, the engine will likely have some downthrust, say a negitive 2. (I'm making these numbers up, btw). In that kind of arrangement, if the plane picks up airspeed due to a dive, the plane will naturally start to pull out.

But if you increase the power, the engine pulling downwards will offset some of the lift of the wing, and the plane won't climb quite so quickly. This kind of arrangements is typical of trainers, lots of positive on the wing, lots of downthrust. When you roll inverted, the downthrust will help you keep the nose up, since it's now upthrust. But the positive angle on the wing and the negitive on the stab are now really working against you. To get a flat bottom wing to produce lift inverted, you have to get the AOA up there fairly high. If the wing is at 2 degrees positive, when you roll inverted, you have to pitch the wing past those 2 degrees, then far enough to get enough lift to maintain level flight. The AOA is controlled by the elevator, so that means you need a lot of down, and the more positive incidence you have on the wing, the more down elevator you will need.

The suggestion above to reflex the ailerons is a good one. Why? Because reflexing the ailerons has the effect of lowering the wing incidence! Drooping the ailerons increased the wing incidence.

So, in summary, to reduce the down elevator needed to fly inverted, do the following:
- move the CG back
- reflex the ailerons or shim up the TE of the wing
- increase down elevator travel

If you are having problems keeping the plane from "falling off" to one side or the other, you can't keep it straight and level while inverted, then you need to look at your dihedral.

With my beater-trainer that I keep around, I can fly full circuits of the field inverted with no problem. I can't quite ouside loop, but I can climb gently inverted. It's a Hobico AirVista, the CG is back a bit, and the elevator is turned up. I'd move the CG back a bit more, but I do instruct on the plane once in a while (when a students plane becomes disabled).