Thanks for extra pictures. Helpful to see more.

First thought I have: consider removing all the vinyl stuff and trying to repaint the door surface.

Do you think this would be possible? If you can't remove, or sand, to create a flat surface to paint, the result would not be pleasing.

Another issue would be how the surface would look being brush-painted instead of sprayed (which is how your cabinets were originally painted).

Maybe, not an ideal fix, but would be a great improvement (compared to what you have now).

If you want to try to repaint, go to a good paint store and ask them to match the cabinet color. If you are hopeful this would work, try repainting the door (maybe painting the door above, too--so those two doors would be a 100% match). If these doors ended up a sightly different color from the color of cabinetry below the counter, the eye might still accept and not notice.

Or, when repainting the one door, re-paint all the cabinetry in the bathroom. That would give the most consistent finish. Not like the factory finish you have now, but consistent and acceptable (if you are patient as a painter)...

My husband painted formerly sprayed-white cabinets in our kitchen, which had become chipped. I'm not sure of technique to have paint stick to that kind of surface, but am sure it can be done and that you can find technique by googling or asking at the paint store.

Re-painting is an art. Takes patience and care, but outcome can be excellent… But, it takes patience (which my husband has and I am lacking)… Our kitchen cabinets turned out beautifully.

Decide if you think repainting is something that would work… If not, here is another thought:

Are the doors on the top cabinet opening and bottom cabinet opening the same size?

If so, as a "quick second idea, " I would move the top undamaged door to the lower space and only have the top cabinet space switch to an open-mode.

If the door on the top is longer than the one on the bottom, I would still see if the top door could be relocated to the bottom space.

... Visually, "open on the top" will be more pleasing than "open on the bottom and closed on the top." (At least that is my opinion---you can remove both doors and have someone hold the undamaged door both above and below and see what you think---Never decide something without trying it out first!)

I think the undamaged door might be relocated even if the natural break between the two cabinets is in a slightly different place. You could readjust shelf locations to make the new door placement "make sense" from the inside...

This might involve moving hinge locations. Is there a handy-man in your life? I don't think this is terribly complicated, but will need a degree of skill to function properly and look okay.

Before adding trim (to switch to an open-shelf concept), you can just remove the door and test the visual of having it as an open space. Contemplate changes before making them permanent...

If "open" drives you crazy, consider finding someone to make a simple (Cheap) frame-of-a-door that you could use in the ABOVE space (having moved the upper door down) with glass (or even a reasonably thick plastic, if glass not safe for the age of your kids). As long as the same width as the rails of the existing door is used and the same routing on the edge, a glass door would be a reasonable match...

Have paint store match new door frame the color to your existing cabinets. Doesn't have to be 100% to fool the eye. The eye expects "matching" and can be much more forgiving that you might think.

Glass door could work… But, you have to feel it would be safe for your children. Adding a glass door is a way of adding a not-exactly-matching door, but safety is, of course, always the priority. Just a thought...

If you are okay with an open space, get some pretty boxes at HomeGoods (or simple white ones at Ikea) to keep the open area pleasant to view (and not so much work to keep up).

A couple large boxes (fabric covered--but not busy-looking-- and stacked--the right scale for the open space) might work very very well… Or, neatly stacked white towels (that no one is allowed to use, or touch, ever : )

These are just ideas. I have a very handy husband (I claim this is just because I am such a skillful nagger) and we have problem-solved all sorts of imperfections caused by accidents. There IS an answer for your problem that will look just fine when you work it out…