I think the one instance where it is great to have mismatched flooring next to each other is between a bathroom and the room next to it. I always want a bathroom to feel disconnected and more private from the more public room that surrounds it.

Yes, you would want to match flooring if they are next to each other so that it is a continuous look.

There are plenty of luxury vinyl tiles that don’t look like wood.
https://www.harveymaria.com/us/

bathroom is a separate room. Put whatever flooring suits the decor/style your plan.

Agree with @Kendrah that I don't mind different flooring in a bathroom - I don't love 2 woods floors that look similar but are a miss - I might go contrasting with the bath floor to make sure the difference looks intentional

I just don't think you can beat tile in a bathroom, unless it's a half bath. If it's a half bath and you want the flooring to match the rest of the basement, buy enough product to cover the whole basement now and have it laid later when you're ready. In a couple years, whatever you buy now probably won't be available anymore, so don't count on getting more later.

Wood floors--real or fake--aren't a great feel in a bathroom. Do a nice tile to get a crisp clean look which is what a basement bathroom needs more than any other bathroom.

(Or put in something cheap and pull it out and replace when you do the rest of the flooring.)

Make the decision whether you want the basement bath to have the same flooring as the rest of the basement. You don't need to pick the flooring now.

If you want the same flooring, then you should follow the advice above and just put in something cheap that you will plan to rip out when you do the rest of the basement. Maybe even a remnant of sheet vinyl would work.

If you want different flooring, then get what you want for the bath. But do heed the advice about not getting to wood-look floors next to each other that are not the same. I would therefore avoid wood-look in the bath since you are leaning that direction for the rest of the basement.