What is the flooring in adjacent rooms? IMO anything you put in those spots that doesn't match exactly is going to stick out like a sore thumb. Those aren't logical locations for any sort of accent. If you have wood in other rooms I would remove all the tile and replace it with matching wood. Planks can be finished on site to match existing. While I am sure this isn't in your budget, a rectangular room (no tricky corners hopefully) isn't a difficult job so the cost shouldn't be extortionate. Is there another layer of tile under the existing?

It’s the same tile throughout the entire house except bedrooms. Underneath appears to be a glossy white tile but it was also damaged in the column demo.

Maybe a faux paint person could paint a tile to closely resemble the rest of your flooring.

Can you try and salvage/reuse some tiles from a closet/pantry/laundry? Then you can use that travertine in the closet to patch the new holes. You might have to try to remove 10 tiles to actually get 2 that you can reuse, but a not quite perfect patch is fine in a closet, but not in a very visible part of your dining/living area.

I’m really appreciating these outside the box ideas!

@Chispa has your best solution.

The only other option I can come up with is to remove more tiles from the dining room, and replace them with something very contrasting. Almost to make a tile rug, when there new tiles are the border of the rug, and keep the old tiles inside and outside the border. But that will be very specific to you, and I wouldn't do it unless you plan to be in this home for a long, long time.

In the photos, it looks like your samples are really close. They might need to be...dirtied. :)

I would use them (if they are real travertine, you can stain them) and then get a big rug under the table. The rug would de-emphasize any mismatch even without covering.