Anything that you do to the shower floor is likely to compromise the waterproofing of the shower.

I agree that the pebble floor in the shower is not a classic look and that it does not match the classic look of the tile on the bathroom floor. The shower tile reads beige; the floor tile reads black/white/grey. It would be less expensive to change the tile on the bathroom floor (outside the shower), than to change the pebble floor in the shower.

Is everything else in the house classic style?

If the current floor is solid & water tight, I don't see why an installer couldn't just mud over the pebble tile and and then install new tiles on top. maybe even the ones you have on the rest of the floor.

The problem is that the walls of the shower look like they were designed to go with the pebbles and not the other finishes of the bathroom. However, that tile can be covered over with new tile also provided the right products are used to adhere the tile.

From the entire shower area, walls and floor, it Appears the shower was created with the idea the rest of the bathroom tile would follow suit. Change bathroom floor tile to complement the shower. It does not have to be a rustic finis. A solid 12 x 24 porcelain tile in the same color family pulled from the pebble floor would also be friendly with the floor outside the bathroom.

As a note of interest, the pebble floor is very similar to the memorial at the American Cemetary in France, which was created from the pebbles on Omaha Beach. My photo is attached.

The wavy shower wall tile looks rather contemporary as compared to your more traditional parts of the bathroom.

I'd get a tall custom made shower curtain that hangs from high up on the ceiling and goes to the floor. Yes, you will see the rustic/modern shower when you are in it, but otherwise, it will not be visible.

I love the white and black floor tile, and the rest of the house is classic. It would cost too much to demo the whole shower, so I think the best next step would be to replace or fix the pebble floors. Maybe even changing the grout so it reads less beige would be better?