I'm sorry to say, but you are going to have to take the mirror down and expose the electrical box. Fixtures are supposed to be mounted to the box. I only hope one is there and they didn't just run wires through the wall without a box. You are not going to be able to cut a mirror in place. A pro has to do that from a template on a firm surface. If the mirror you have is mounted with mastic instead of clips, too much mastic will be on the back for a pro to cut it safely. It would have good potential to shatter. It would need to be replaced. If it's just mounted with clips, it might be saved and altered. Either way, the box has to be exposed so the light can successfully be mounted to it.

How many lumins will your fixture put out? I just did 2 bathrooms over that last 3 years. In one bathroom the light is great and my lumins = 3000 for my 6.5x9 foot room, with a nice sized window and whitish decor. Even at night it is nice and bright. In the other bathroom that is about 8x9.5 feet I have 6400 lumins that I calculate and it is a little dimmer than the other BR. The room is a bit darker with walnut trim and doors, so that might have something to do with it. I would go for light other than the type of light if you can. The vanity light isn't on all that much during the day, but you will want bright light for makeup and shaving. To clarify the last paragraph. I would try to get the lumins I need, rather then only looking at the type of light. Because the vanity light isn't on but a few minutes a day, and bright light at the vanity is a must.

You should give LED strip lights a try. I use them under the cupboard in the kitchen, and I installed them myself, which was not hard at all. I got mine from https://www.ukled.co.uk/led-strip-lights.html, and I highly recommend them. I chose the waterproof ones since I don't want them to get ruined if I spill anything on them since you never know what can happen in the kitchen. I use them for architectural and landscape lighting effects as well. I put them in my backyard and terrace, and they look phenomenal. You won't believe how many compliments I got for the decor!

I think you'll want to buy two mirrors and three sconces. The balance will work out much better being able to have one above each sink. Furthermore, I would propose getting three lights as that will probably help with ensuring you have more evenly spread lighting to work with in the mirrors. I realize this bathroom vanity is longer but it does have the two mirrors and three lights. Definitely our preference when building.