Sorry but to lighten up the basement remove that oppresive looking thing above the bar and add some nice more colorful not wood stools. Then do the floor as planned and the wall color

Sounds like a great idea. Light flooring next to cherry cabinetry will look great, nice contrast and will help to lighten. Even lighter with a White Birch or Maple.

Look for something with very little grain lines. a tight grain.

I am wondering if Pale Oak is light enough will have a greige/gray undertone.

How about a sample of SW Dover White? which will help bounce light around the space, and if you need a little color, you could paint the TV wall a Navy Blue for interest. Or a hunter green like the table top.Polished Brass fixtures for a men's club vibe.

no tree in a pool hall. or at least put it by the window.

Make sure your sub floor is very even etc or pay for really good prep - carpet is often used in basements when the floor is not super flat as its really forgiving. SO many LVP fails come back to bad sub floors and cheap installs

The light colored floor looks grand in that space. I agree with whoever said that the heavy wood piece on the ceiling seems oppresive. I would opt to use that lovely space for something other than a bar. It could be stunning.