I discovered I had not thoroughly cut through all the caulking between the flange of the door frame and the side of the motorhome, so completed that. Using a large C clamp, a metal bar, and some wood pieces, I was able to force the left side away from the motorhome side, which started to loosen it. Prying on the flange of the right side broke it free. The hole was tight fit for the door, which is what made it so hard.
In the pictures, you can see the hole is framed with aluminum tubes on the top and sides, but not the bottom. That door was not going to fall out! I think cutting into the upper horizontal aluminum piece will weaken the structure, so I'll butt the new door up against it. The vertical aluminum piece on the left side will have to be cut off, as the new door is twice as wide. It's also an inch taller, so I'll have to cut into the wood at the bottom.
So far, it looks like everything will still fit, though I'm disappointed I can't mount the new door a bit higher.