You are correct- fake wood floors look incredibly fake next to real wood. The first thing I would do is paint the paneling and then choose the most neutral lvp you can find. The paneling is just birch plywood- it’s nothing rare or unique (I’m trying to assuage your guilt before you paint it) If you insist on keeping it don’t try to match it. Get a light mid tone ‘European oak’ so there is contrast.

Your paneling is off the charts beautiful. The idea of ugly-ing it down to match fake copy flooring is ridiculous. Find a tile that's not wood like. Easy.

Also get rid of the popcorn finish on the ceiling or at least paint the ceiling with a flat white color. The glare/reflection from the windows, on the ceiling, is awful.

Although it is not as easy to find they make some relatively plain, neutral stone-look luxury vinyl tile.

These are Mannington Adura

This is Karndean Ivory Riven Slate

This is Armstrong Terra, with a terrazzo kind of appearance:

I have always liked the slate or slate look flooring with wood paneling.

Maria Killam has a post where she shows some great matches of LVT to wood floors - we've come a long way in the look of LVP, so I wouldn't just say it all looks fake next to the real thing.

I don't think I would want wood look LVP with your wood paneling, but it could be done if that is what you like.

I agree with palimpsest that you could find some nice tile or stone look LVT.

I think this might be nice with your walls

or possibly this: