Yes moving a beam (if possible) can be a big expense. I wouldn't be too worried about closing this off. what is the wall plate covering? Probably a junction box for some electrical that is ran in your bathroom or somewhere else which you're not supposed to cover so maybe you can relocate.

The idea of closing off the ceiling to even out with the soffit can be a good idea and you can also add lighting. I wouldn't drop it everywhere but over the vanity will be okay.

If that wall plate is covering an electrical junction box, then you can't close the area in. Junction boxes need to be left accessible.

How about ceiling panels? The simplest, ugliest example is the ceilings you see in every big box store from Walmart to Home Cheapo.

Use that idea, but instead of a metal grid, use wood. Instead of fiberglass panels, use plywood. It's simple carpentry, nothing difficult, and could be as sleek or ornate as you want. To access wiring, just lift out a panel. Any sort of lighting could be used.

What other changes are you trying to accomplish? What is the plan for the rest of the room?

Yes, I would close it off. Since the suspected junction box behind the wall plate doesn't have to move very far, there just may be enough slack in the wiring to relocate it to the new dropped ceiling.

I would put up some lighting.

Oh, too funny! You just got rid of what I suggested you to put back!