Changing the flooring is a great idea to draw attention away from the warm color tile. It sounds like you want a more cooler feel but are having a problem trying to incorporate the warmer existing tile. Have you ever looked into painting it? There are epoxy refinishing kits you can you to paint the tile and grout white ! Very cost effective. In addition to that, you can also change the trim kit in the shower and tub and that will make it look updated as well. You definitely have a sense on how to blend them, if you don't want to refinish the tile, with the pictures you have above. You do have a good balance of warm and cool elements to blend the two and at minimum I would at least change the trim kit on your tub and shower so they don't look outdated with the newer items. I just remodeled a home on a budget where I had the same situation (keeping some items warm and updating with cool) and that may give you some more ideas on how to play with the two elements and achieve an over all balance. Hope this helps !

I think changing the trim kit would be the best solution to bring it up to date and not look outdated.