It's a pretty clever idea and a solid question that unseen mass might cause gravity outside the observable that tugs on the universe and might be the cause of dark energy as opposed to some unknown energy that seems to permeate all space. It's not a new idea, but it's worth answering.
There's 2, possibly 3 (or 4) pretty big problems to that approach. The first is that Dark Energy expansion is very uniform. For it to be explained by gravity, that gravity would need to be close to uniform around the observable universe, with Earth close to the center. A giant "edge" as you called it, or maybe we're in a low mass bubble surrounded by higher mass. Not impossible, but somewhat improbable that the Earth is in the center of this theoretical gravity field pulling the observable universe apart.
The 2nd problem is time. Gravity travels at the speed of light, so this gravity field would need to have existed prior to the big bang or be very close to the edge of the observable universe. Still not impossible, but it's a difficult theory to work with as it requires a few leaps of faith.
A possible 3rd problem is that measurements show that the universe is very flat. I don't know if a gravity field of the sort you suggest would be flat or not, I suspect not, unless it was enormously large.
Given enough time, maybe they can find a test to detect gravitons (if they exist) or, find dark energy and answer this one with better certainty. The question is asked here on Quora if interested.
I'll add a 4th problem. The velocity of observed young galaxies at the edge of the observable universe is close to the speed of light away from us in all directions. Relative velocity that high is difficult to explain by high mass outside of the observable universe cause that mass would need to be enormous, measurably greater than the mass of the observable universe. All the factors taken into account, it's an improbable explanation
If I've missed anything obvious or broken any laws of physics with the answer above, corrections are always welcome.