It's NEVER a good idea to put wood look tile/vinyl next to real wood. I agree with above. You need to get an experienced plumber in to address the drain issue. Moisture in walls causes mold and all kinds of health problems.

We have had plumbers come out. It is vent pipe on the top of our roof that gets clogged. We live in a very wooded area with high trees.

We are planning on replacing all the wood floor.

Call a good plumber. Get the shower fixed.

I would get the flooding issue sorted before replacing floors.

Then I would put LVP in. But only assuming it doesn’t meet up with the real wood.

You need a vent that is high enough to clear whatever happens on the roof, that is where you start BTW if the water floods the floor no matter what you have there the subfloor still gets wet and needs to be replaced.Wood look tile is already becoming a dated look and living with all tile floors is not pleasant on knees and feet. Fix the plumbing then replace your wood floors have them finished to match the rest , that IMO is the best way to handle this.